diff --git a/pydantic/fields.py b/pydantic/fields.py
--- a/pydantic/fields.py
+++ b/pydantic/fields.py
@@ -34,7 +34,6 @@
     Callable,
     ForwardRef,
     NoArgAnyCallable,
-    NoneType,
     display_as_type,
     get_args,
     get_origin,
@@ -563,10 +562,11 @@ def _type_analysis(self) -> None:  # noqa: C901 (ignore complexity)
         if is_union(origin):
             types_ = []
             for type_ in get_args(self.type_):
-                if type_ is NoneType:
+                if is_none_type(type_) or type_ is Any or type_ is object:
                     if self.required is Undefined:
                         self.required = False
                     self.allow_none = True
+                if is_none_type(type_):
                     continue
                 types_.append(type_)
 

`allow_none` attribute for field `a` remains `False`. Possible fix may be in pydantic/fields.py:566:
```
if is_union_origin(origin):
    types_ = []
    for type_ in get_args(self.type_):
        if type_ is NoneType or type_ is Any or type_ is object:
            if self.required is Undefined:
                self.required = False
            self.allow_none = True
            continue
        types_.append(type_)
```
@mykhailoleskiv That almost works, except it doesn't add `Any` or `object` to `types_` (which eventually becomes `ModelField.sub_fields`). So `M(a="abcd")` or something like that no longer works. I'll open a PR
diff --git a/tests/test_edge_cases.py b/tests/test_edge_cases.py
--- a/tests/test_edge_cases.py
+++ b/tests/test_edge_cases.py
@@ -83,6 +83,23 @@ class Model(BaseModel):
     ]
 
 
+def test_union_int_any():
+    class Model(BaseModel):
+        v: Union[int, Any]
+
+    m = Model(v=123)
+    assert m.v == 123
+
+    m = Model(v='123')
+    assert m.v == 123
+
+    m = Model(v='foobar')
+    assert m.v == 'foobar'
+
+    m = Model(v=None)
+    assert m.v is None
+
+
 def test_union_priority():
     class ModelOne(BaseModel):
         v: Union[int, str] = ...

"none is not an allowed value" when using Union with Any
### Checks

* [x] I added a descriptive title to this issue
* [x] I have searched (google, github) for similar issues and couldn't find anything
* [x] I have read and followed [the docs](https://pydantic-docs.helpmanual.io/) and still think this is a bug

# Bug
It seems that `None` is allowed when a field's type is `Any` (as is expected, stated in the docs), but `None` is not allowed when a field's type is `Union[SomeType, Any]`.

Output of `python -c "import pydantic.utils; print(pydantic.utils.version_info())"`:
```
             pydantic version: 1.8.2
            pydantic compiled: True
                 install path: C:\Users\TobyHarradine\PycharmProjects\orchestration\.venv\Lib\site-packages\pydantic
               python version: 3.7.9 (tags/v3.7.9:13c94747c7, Aug 17 2020, 18:58:18) [MSC v.1900 64 bit (AMD64)]
                     platform: Windows-10-10.0.19041-SP0
     optional deps. installed: ['typing-extensions']
```

Code which reproduces the issue:
```pycon
>>> import pydantic
>>> class M(pydantic.BaseModel):
...     a: Any
...
>>> M(a=1)
M(a=1)
>>> M(a=None)
M(a=None)
>>> class M(pydantic.BaseModel):
...     a: Union[int, Any]
...
>>> M(a=1)
M(a=1)
>>> M(a="abcd")
M(a='abcd')
>>> M(a=None)
Traceback (most recent call last):
  File "<input>", line 1, in <module>
  File "pydantic\main.py", line 406, in pydantic.main.BaseModel.__init__
pydantic.error_wrappers.ValidationError: 1 validation error for M
a
  none is not an allowed value (type=type_error.none.not_allowed)

```

This may be somewhat related to #1624.

tests/test_edge_cases.py::test_union_int_any
tests/test_edge_cases.py::test_str_bytes tests/test_edge_cases.py::test_str_bytes_none tests/test_edge_cases.py::test_union_int_str tests/test_edge_cases.py::test_union_priority tests/test_edge_cases.py::test_typed_list tests/test_edge_cases.py::test_typed_set tests/test_edge_cases.py::test_dict_dict tests/test_edge_cases.py::test_none_list tests/test_edge_cases.py::test_typed_dict[value0-result0] tests/test_edge_cases.py::test_typed_dict[value1-result1] tests/test_edge_cases.py::test_typed_dict[value2-result2] tests/test_edge_cases.py::test_typed_dict_error[1-errors0] tests/test_edge_cases.py::test_typed_dict_error[value1-errors1] tests/test_edge_cases.py::test_typed_dict_error[value2-errors2] tests/test_edge_cases.py::test_dict_key_error tests/test_edge_cases.py::test_tuple tests/test_edge_cases.py::test_tuple_more tests/test_edge_cases.py::test_tuple_length_error tests/test_edge_cases.py::test_tuple_invalid tests/test_edge_cases.py::test_tuple_value_error tests/test_edge_cases.py::test_recursive_list tests/test_edge_cases.py::test_recursive_list_error tests/test_edge_cases.py::test_list_unions tests/test_edge_cases.py::test_recursive_lists tests/test_edge_cases.py::test_str_enum tests/test_edge_cases.py::test_any_dict tests/test_edge_cases.py::test_success_values_include tests/test_edge_cases.py::test_include_exclude_unset tests/test_edge_cases.py::test_include_exclude_defaults tests/test_edge_cases.py::test_skip_defaults_deprecated tests/test_edge_cases.py::test_advanced_exclude tests/test_edge_cases.py::test_advanced_exclude_by_alias tests/test_edge_cases.py::test_advanced_value_include tests/test_edge_cases.py::test_advanced_value_exclude_include tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude0-expected0] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude1-expected1] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude2-expected2] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude3-expected3] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude4-expected4] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude5-expected5] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude6-expected6] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude7-expected7] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude8-expected8] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude9-expected9] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude10-expected10] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude11-expected11] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude12-expected12] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude13-expected13] tests/test_edge_cases.py::test_advanced_exclude_nested_lists[exclude14-expected14] tests/test_edge_cases.py::test_advanced_include_nested_lists[include0-expected0] tests/test_edge_cases.py::test_advanced_include_nested_lists[include1-expected1] tests/test_edge_cases.py::test_advanced_include_nested_lists[include2-expected2] tests/test_edge_cases.py::test_advanced_include_nested_lists[include3-expected3] tests/test_edge_cases.py::test_advanced_include_nested_lists[include4-expected4] tests/test_edge_cases.py::test_advanced_include_nested_lists[include5-expected5] tests/test_edge_cases.py::test_advanced_include_nested_lists[include6-expected6] tests/test_edge_cases.py::test_advanced_include_nested_lists[include7-expected7] tests/test_edge_cases.py::test_advanced_include_nested_lists[include8-expected8] tests/test_edge_cases.py::test_advanced_include_nested_lists[include9-expected9] tests/test_edge_cases.py::test_advanced_include_nested_lists[include10-expected10] tests/test_edge_cases.py::test_advanced_include_nested_lists[include11-expected11] tests/test_edge_cases.py::test_advanced_include_nested_lists[include12-expected12] tests/test_edge_cases.py::test_advanced_include_nested_lists[include13-expected13] tests/test_edge_cases.py::test_advanced_include_nested_lists[include14-expected14] tests/test_edge_cases.py::test_field_set_ignore_extra tests/test_edge_cases.py::test_field_set_allow_extra tests/test_edge_cases.py::test_field_set_field_name tests/test_edge_cases.py::test_values_order tests/test_edge_cases.py::test_inheritance tests/test_edge_cases.py::test_inheritance_subclass_default tests/test_edge_cases.py::test_invalid_type tests/test_edge_cases.py::test_valid_string_types[a tests/test_edge_cases.py::test_valid_string_types[some tests/test_edge_cases.py::test_valid_string_types[value2-foobar] tests/test_edge_cases.py::test_valid_string_types[123-123] tests/test_edge_cases.py::test_valid_string_types[123.45-123.45] tests/test_edge_cases.py::test_valid_string_types[value5-12.45] tests/test_edge_cases.py::test_valid_string_types[True-True] tests/test_edge_cases.py::test_valid_string_types[False-False] tests/test_edge_cases.py::test_valid_string_types[a10-a10] tests/test_edge_cases.py::test_valid_string_types[whatever-whatever] tests/test_edge_cases.py::test_invalid_string_types[value0-errors0] tests/test_edge_cases.py::test_invalid_string_types[value1-errors1] tests/test_edge_cases.py::test_inheritance_config tests/test_edge_cases.py::test_partial_inheritance_config tests/test_edge_cases.py::test_annotation_inheritance tests/test_edge_cases.py::test_string_none tests/test_edge_cases.py::test_return_errors_ok tests/test_edge_cases.py::test_return_errors_error tests/test_edge_cases.py::test_optional_required tests/test_edge_cases.py::test_invalid_validator tests/test_edge_cases.py::test_unable_to_infer tests/test_edge_cases.py::test_multiple_errors tests/test_edge_cases.py::test_validate_all tests/test_edge_cases.py::test_force_extra tests/test_edge_cases.py::test_illegal_extra_value tests/test_edge_cases.py::test_multiple_inheritance_config tests/test_edge_cases.py::test_submodel_different_type tests/test_edge_cases.py::test_self tests/test_edge_cases.py::test_self_recursive[BaseModel] tests/test_edge_cases.py::test_self_recursive[BaseSettings] tests/test_edge_cases.py::test_nested_init[BaseModel] tests/test_edge_cases.py::test_nested_init[BaseSettings] tests/test_edge_cases.py::test_init_inspection tests/test_edge_cases.py::test_type_on_annotation tests/test_edge_cases.py::test_assign_type tests/test_edge_cases.py::test_optional_subfields tests/test_edge_cases.py::test_not_optional_subfields tests/test_edge_cases.py::test_optional_field_constraints tests/test_edge_cases.py::test_field_str_shape tests/test_edge_cases.py::test_field_type_display[int-int] tests/test_edge_cases.py::test_field_type_display[type_1-Optional[int]] tests/test_edge_cases.py::test_field_type_display[type_2-Union[NoneType, tests/test_edge_cases.py::test_field_type_display[type_3-Union[int, tests/test_edge_cases.py::test_field_type_display[type_4-List[int]] tests/test_edge_cases.py::test_field_type_display[type_5-Tuple[int, tests/test_edge_cases.py::test_field_type_display[type_6-Union[List[int], tests/test_edge_cases.py::test_field_type_display[type_7-List[Tuple[int, tests/test_edge_cases.py::test_field_type_display[type_8-Mapping[int, tests/test_edge_cases.py::test_field_type_display[type_9-FrozenSet[int]] tests/test_edge_cases.py::test_field_type_display[type_10-Tuple[int, tests/test_edge_cases.py::test_field_type_display[type_11-Optional[List[int]]] tests/test_edge_cases.py::test_field_type_display[dict-dict] tests/test_edge_cases.py::test_field_type_display[type_13-DisplayGen[bool, tests/test_edge_cases.py::test_any_none tests/test_edge_cases.py::test_type_var_any tests/test_edge_cases.py::test_type_var_constraint tests/test_edge_cases.py::test_type_var_bound tests/test_edge_cases.py::test_dict_bare tests/test_edge_cases.py::test_list_bare tests/test_edge_cases.py::test_dict_any tests/test_edge_cases.py::test_modify_fields tests/test_edge_cases.py::test_exclude_none tests/test_edge_cases.py::test_exclude_none_recursive tests/test_edge_cases.py::test_exclude_none_with_extra tests/test_edge_cases.py::test_str_method_inheritance tests/test_edge_cases.py::test_repr_method_inheritance tests/test_edge_cases.py::test_optional_validator tests/test_edge_cases.py::test_required_optional tests/test_edge_cases.py::test_required_any tests/test_edge_cases.py::test_custom_generic_validators tests/test_edge_cases.py::test_custom_generic_arbitrary_allowed tests/test_edge_cases.py::test_custom_generic_disallowed tests/test_edge_cases.py::test_hashable_required tests/test_edge_cases.py::test_hashable_optional[1] tests/test_edge_cases.py::test_hashable_optional[None] tests/test_edge_cases.py::test_default_factory_called_once tests/test_edge_cases.py::test_default_factory_validator_child tests/test_edge_cases.py::test_cython_function_untouched tests/test_edge_cases.py::test_resolve_annotations_module_missing tests/test_edge_cases.py::test_iter_coverage tests/test_edge_cases.py::test_config_field_info tests/test_edge_cases.py::test_config_field_info_alias tests/test_edge_cases.py::test_config_field_info_merge tests/test_edge_cases.py::test_config_field_info_allow_mutation tests/test_edge_cases.py::test_arbitrary_types_allowed_custom_eq