diff --git a/moto/s3/models.py b/moto/s3/models.py
index 98229539e..5fa115d69 100644
--- a/moto/s3/models.py
+++ b/moto/s3/models.py
@@ -169,6 +169,9 @@ class FakeKey(BaseModel):
             raise InvalidStorageClass(storage=storage)
         self._storage_class = storage
 
+    def set_expiry(self, expiry):
+        self._expiry = expiry
+
     def set_acl(self, acl):
         self.acl = acl
 
@@ -1689,6 +1692,9 @@ class S3Backend(BaseBackend):
             new_key.set_storage_class(storage)
         if acl is not None:
             new_key.set_acl(acl)
+        if key.storage_class in "GLACIER":
+            # Object copied from Glacier object should not have expiry
+            new_key.set_expiry(None)
 
         dest_bucket.keys[dest_key_name] = new_key
 
diff --git a/moto/s3/responses.py b/moto/s3/responses.py
index 530365a6e..a7fb82210 100644
--- a/moto/s3/responses.py
+++ b/moto/s3/responses.py
@@ -1276,7 +1276,13 @@ class ResponseObject(_TemplateEnvironmentMixin, ActionAuthenticatorMixin):
 
             if key is not None:
                 if key.storage_class in ["GLACIER", "DEEP_ARCHIVE"]:
-                    raise ObjectNotInActiveTierError(key)
+                    if key.response_dict.get(
+                        "x-amz-restore"
+                    ) is None or 'ongoing-request="true"' in key.response_dict.get(
+                        "x-amz-restore"
+                    ):
+                        raise ObjectNotInActiveTierError(key)
+
                 self.backend.copy_key(
                     src_bucket,
                     src_key,
diff --git a/tests/test_s3/test_s3_storageclass.py b/tests/test_s3/test_s3_storageclass.py
index f1a0479b2..a89b4a896 100644
--- a/tests/test_s3/test_s3_storageclass.py
+++ b/tests/test_s3/test_s3_storageclass.py
@@ -156,7 +156,7 @@ def test_s3_default_storage_class():
 
 
 @mock_s3
-def test_s3_copy_object_error_for_glacier_storage_class():
+def test_s3_copy_object_error_for_glacier_storage_class_not_restored():
     s3 = boto3.client("s3")
     s3.create_bucket(
         Bucket="Bucket", CreateBucketConfiguration={"LocationConstraint": "us-west-1"}
@@ -177,7 +177,7 @@ def test_s3_copy_object_error_for_glacier_storage_class():
 
 
 @mock_s3
-def test_s3_copy_object_error_for_deep_archive_storage_class():
+def test_s3_copy_object_error_for_deep_archive_storage_class_not_restored():
     s3 = boto3.client("s3")
     s3.create_bucket(
         Bucket="Bucket", CreateBucketConfiguration={"LocationConstraint": "us-west-1"}
@@ -195,3 +195,57 @@ def test_s3_copy_object_error_for_deep_archive_storage_class():
         )
 
     exc.exception.response["Error"]["Code"].should.equal("ObjectNotInActiveTierError")
+
+
+@mock_s3
+def test_s3_copy_object_for_glacier_storage_class_restored():
+    s3 = boto3.client("s3", region_name="us-east-1")
+    s3.create_bucket(Bucket="Bucket")
+
+    s3.put_object(
+        Bucket="Bucket", Key="First_Object", Body="Body", StorageClass="GLACIER"
+    )
+
+    s3.create_bucket(Bucket="Bucket2")
+    s3.restore_object(Bucket="Bucket", Key="First_Object", RestoreRequest={"Days": 123})
+
+    s3.copy_object(
+        CopySource={"Bucket": "Bucket", "Key": "First_Object"},
+        Bucket="Bucket2",
+        Key="Second_Object",
+    )
+
+    list_of_copied_objects = s3.list_objects(Bucket="Bucket2")
+    # checks that copy of restored Glacier object has STANDARD storage class
+    list_of_copied_objects["Contents"][0]["StorageClass"].should.equal("STANDARD")
+    # checks that metadata of copy has no Restore property
+    s3.head_object(Bucket="Bucket2", Key="Second_Object").should.not_have.property(
+        "Restore"
+    )
+
+
+@mock_s3
+def test_s3_copy_object_for_deep_archive_storage_class_restored():
+    s3 = boto3.client("s3", region_name="us-east-1")
+    s3.create_bucket(Bucket="Bucket")
+
+    s3.put_object(
+        Bucket="Bucket", Key="First_Object", Body="Body", StorageClass="DEEP_ARCHIVE"
+    )
+
+    s3.create_bucket(Bucket="Bucket2")
+    s3.restore_object(Bucket="Bucket", Key="First_Object", RestoreRequest={"Days": 123})
+
+    s3.copy_object(
+        CopySource={"Bucket": "Bucket", "Key": "First_Object"},
+        Bucket="Bucket2",
+        Key="Second_Object",
+    )
+
+    list_of_copied_objects = s3.list_objects(Bucket="Bucket2")
+    # checks that copy of restored Glacier object has STANDARD storage class
+    list_of_copied_objects["Contents"][0]["StorageClass"].should.equal("STANDARD")
+    # checks that metadata of copy has no Restore property
+    s3.head_object(Bucket="Bucket2", Key="Second_Object").should.not_have.property(
+        "Restore"
+    )