diff --git a/python/cudf/cudf/core/index.py b/python/cudf/cudf/core/index.py
index 324db416be4..cd882aba297 100644
--- a/python/cudf/cudf/core/index.py
+++ b/python/cudf/cudf/core/index.py
@@ -1,4 +1,4 @@
-# Copyright (c) 2018-2022, NVIDIA CORPORATION.
+# Copyright (c) 2018-2023, NVIDIA CORPORATION.
 
 from __future__ import annotations
 
@@ -313,9 +313,20 @@ def copy(self, name=None, deep=False, dtype=None, names=None):
             Ignored for RangeIndex
         dtype : numpy dtype optional (default: None)
             Target dtype for underlying range data
+
+            .. deprecated:: 23.02
+
+               The `dtype` parameter is deprecated and will be removed in
+               a future version of cudf. Use the `astype` method instead.
+
         names : list-like optional (default: False)
             Kept compatibility with MultiIndex. Should not be used.
 
+            .. deprecated:: 23.04
+
+               The parameter `names` is deprecated and will be removed in
+               a future version of cudf. Use the `name` parameter instead.
+
         Returns
         -------
         New RangeIndex instance with same range, casted to new dtype
@@ -327,6 +338,13 @@ def copy(self, name=None, deep=False, dtype=None, names=None):
                 FutureWarning,
             )
 
+        if names is not None:
+            warnings.warn(
+                "parameter names is deprecated and will be removed in a "
+                "future version. Use the name parameter instead.",
+                FutureWarning,
+            )
+
         dtype = self.dtype if dtype is None else dtype
 
         if not np.issubdtype(dtype, np.signedinteger):
@@ -1135,9 +1153,20 @@ def copy(self, name=None, deep=False, dtype=None, names=None):
             With ``deep=False`` the original data is used
         dtype : numpy dtype, default None
             Target datatype to cast into, use original dtype when None
+
+            .. deprecated:: 23.02
+
+               The `dtype` parameter is deprecated and will be removed in
+               a future version of cudf. Use the `astype` method instead.
+
         names : list-like, default False
             Kept compatibility with MultiIndex. Should not be used.
 
+            .. deprecated:: 23.04
+
+               The parameter `names` is deprecated and will be removed in
+               a future version of cudf. Use the `name` parameter instead.
+
         Returns
         -------
         New index instance, casted to new dtype
@@ -1149,6 +1178,13 @@ def copy(self, name=None, deep=False, dtype=None, names=None):
                 FutureWarning,
             )
 
+        if names is not None:
+            warnings.warn(
+                "parameter names is deprecated and will be removed in a "
+                "future version. Use the name parameter instead.",
+                FutureWarning,
+            )
+
         dtype = self.dtype if dtype is None else dtype
         name = self.name if name is None else name
 
diff --git a/python/cudf/cudf/core/multiindex.py b/python/cudf/cudf/core/multiindex.py
index 783c3996400..573a3f7f1d7 100644
--- a/python/cudf/cudf/core/multiindex.py
+++ b/python/cudf/cudf/core/multiindex.py
@@ -336,11 +336,29 @@ def copy(
             Names for each of the index levels.
         dtype : object, optional (default None)
             MultiIndex dtype, only supports None or object type
+
+            .. deprecated:: 23.02
+
+               The `dtype` parameter is deprecated and will be removed in
+               a future version of cudf. Use the `astype` method instead.
+
         levels : sequence of arrays, optional (default None)
             The unique labels for each level. Original values used if None.
+
+            .. deprecated:: 23.02
+
+               The `levels` parameter is deprecated and will be removed in
+               a future version of cudf.
+
         codes : sequence of arrays, optional (default None)
             Integers for each level designating which label at each location.
             Original values used if None.
+
+            .. deprecated:: 23.02
+
+               The `codes` parameter is deprecated and will be removed in
+               a future version of cudf.
+
         deep : Bool (default False)
             If True, `._data`, `._levels`, `._codes` will be copied. Ignored if
             `levels` or `codes` are specified.
@@ -401,6 +419,13 @@ def copy(
                 FutureWarning,
             )
 
+        if dtype is not None:
+            warnings.warn(
+                "parameter dtype is deprecated and will be removed in a "
+                "future version. Use the astype method instead.",
+                FutureWarning,
+            )
+
         dtype = object if dtype is None else dtype
         if not pd.core.dtypes.common.is_object_dtype(dtype):
             raise TypeError("Dtype for MultiIndex only supports object type.")