diff --git a/django/db/backends/postgresql/schema.py b/django/db/backends/postgresql/schema.py
index f6be40f846..8d4db4ea1c 100644
--- a/django/db/backends/postgresql/schema.py
+++ b/django/db/backends/postgresql/schema.py
@@ -104,6 +104,15 @@ class DatabaseSchemaEditor(BaseDatabaseSchemaEditor):
 
     def _alter_field(self, model, old_field, new_field, old_type, new_type,
                      old_db_params, new_db_params, strict=False):
+        # Drop indexes on varchar/text columns that are changing to a different
+        # type.
+        if (old_field.db_index or old_field.unique) and (
+            (old_type.startswith('varchar') and not new_type.startswith('varchar')) or
+            (old_type.startswith('text') and not new_type.startswith('text'))
+        ):
+            index_name = self._create_index_name(model, [old_field.column], suffix='_like')
+            self.execute(self._delete_constraint_sql(self.sql_delete_index, model, index_name))
+
         super()._alter_field(
             model, old_field, new_field, old_type, new_type, old_db_params,
             new_db_params, strict,
diff --git a/tests/schema/models.py b/tests/schema/models.py
index 0e04dbab87..4512d8bc01 100644
--- a/tests/schema/models.py
+++ b/tests/schema/models.py
@@ -17,6 +17,13 @@ class Author(models.Model):
         apps = new_apps
 
 
+class AuthorCharFieldWithIndex(models.Model):
+    char_field = models.CharField(max_length=31, db_index=True)
+
+    class Meta:
+        apps = new_apps
+
+
 class AuthorTextFieldWithIndex(models.Model):
     text_field = models.TextField(db_index=True)
 
diff --git a/tests/schema/tests.py b/tests/schema/tests.py
index ef4b192a36..293ae8e546 100644
--- a/tests/schema/tests.py
+++ b/tests/schema/tests.py
@@ -29,11 +29,11 @@ from .fields import (
     CustomManyToManyField, InheritedManyToManyField, MediumBlobField,
 )
 from .models import (
-    Author, AuthorTextFieldWithIndex, AuthorWithDefaultHeight,
-    AuthorWithEvenLongerName, AuthorWithIndexedName, Book, BookForeignObj,
-    BookWeak, BookWithLongName, BookWithO2O, BookWithoutAuthor, BookWithSlug,
-    IntegerPK, Node, Note, NoteRename, Tag, TagIndexed, TagM2MTest,
-    TagUniqueRename, Thing, UniqueTest, new_apps,
+    Author, AuthorCharFieldWithIndex, AuthorTextFieldWithIndex,
+    AuthorWithDefaultHeight, AuthorWithEvenLongerName, AuthorWithIndexedName,
+    Book, BookForeignObj, BookWeak, BookWithLongName, BookWithO2O,
+    BookWithoutAuthor, BookWithSlug, IntegerPK, Node, Note, NoteRename, Tag,
+    TagIndexed, TagM2MTest, TagUniqueRename, Thing, UniqueTest, new_apps,
 )
 
 
@@ -49,9 +49,10 @@ class SchemaTests(TransactionTestCase):
     available_apps = []
 
     models = [
-        Author, AuthorWithDefaultHeight, AuthorWithEvenLongerName, Book,
-        BookWeak, BookWithLongName, BookWithO2O, BookWithSlug, IntegerPK, Node,
-        Note, Tag, TagIndexed, TagM2MTest, TagUniqueRename, Thing, UniqueTest,
+        Author, AuthorCharFieldWithIndex, AuthorTextFieldWithIndex,
+        AuthorWithDefaultHeight, AuthorWithEvenLongerName, Book, BookWeak,
+        BookWithLongName, BookWithO2O, BookWithSlug, IntegerPK, Node, Note,
+        Tag, TagIndexed, TagM2MTest, TagUniqueRename, Thing, UniqueTest,
     ]
 
     # Utility functions
@@ -222,6 +223,49 @@ class SchemaTests(TransactionTestCase):
         else:
             self.fail("No FK constraint for author_id found")
 
+    @skipUnlessDBFeature('supports_foreign_keys')
+    def test_char_field_with_db_index_to_fk(self):
+        # Create the table
+        with connection.schema_editor() as editor:
+            editor.create_model(Author)
+            editor.create_model(AuthorCharFieldWithIndex)
+        # Change CharField to FK
+        old_field = AuthorCharFieldWithIndex._meta.get_field('char_field')
+        new_field = ForeignKey(Author, CASCADE, blank=True)
+        new_field.set_attributes_from_name('char_field')
+        with connection.schema_editor() as editor:
+            editor.alter_field(AuthorCharFieldWithIndex, old_field, new_field, strict=True)
+        # The new FK constraint is present
+        constraints = self.get_constraints(AuthorCharFieldWithIndex._meta.db_table)
+        constraint_fk = None
+        for name, details in constraints.items():
+            if details['columns'] == ['char_field_id'] and details['foreign_key']:
+                constraint_fk = details['foreign_key']
+                break
+        self.assertEqual(constraint_fk, ('schema_author', 'id'))
+
+    @skipUnlessDBFeature('supports_foreign_keys')
+    @skipUnlessDBFeature('supports_index_on_text_field')
+    def test_text_field_with_db_index_to_fk(self):
+        # Create the table
+        with connection.schema_editor() as editor:
+            editor.create_model(Author)
+            editor.create_model(AuthorTextFieldWithIndex)
+        # Change TextField to FK
+        old_field = AuthorTextFieldWithIndex._meta.get_field('text_field')
+        new_field = ForeignKey(Author, CASCADE, blank=True)
+        new_field.set_attributes_from_name('text_field')
+        with connection.schema_editor() as editor:
+            editor.alter_field(AuthorTextFieldWithIndex, old_field, new_field, strict=True)
+        # The new FK constraint is present
+        constraints = self.get_constraints(AuthorTextFieldWithIndex._meta.db_table)
+        constraint_fk = None
+        for name, details in constraints.items():
+            if details['columns'] == ['text_field_id'] and details['foreign_key']:
+                constraint_fk = details['foreign_key']
+                break
+        self.assertEqual(constraint_fk, ('schema_author', 'id'))
+
     @skipUnlessDBFeature('supports_foreign_keys')
     def test_fk_to_proxy(self):
         "Creating a FK to a proxy model creates database constraints."