diff --git a/pandas/core/missing.py b/pandas/core/missing.py
index 1a07b5614eb38..abf848d7070f4 100644
--- a/pandas/core/missing.py
+++ b/pandas/core/missing.py
@@ -430,7 +430,7 @@ def _interpolate_1d(
     if method in NP_METHODS:
         # np.interp requires sorted X values, #21037
 
-        indexer = np.argsort(indices[valid])
+        indexer = np.argsort(indices[valid], kind="stable")
         result[invalid] = np.interp(
             indices[invalid], indices[valid][indexer], yvalues[valid][indexer]
         )
diff --git a/pandas/tests/series/methods/test_interpolate.py b/pandas/tests/series/methods/test_interpolate.py
index 8ca2d37016691..8a5961f06a498 100644
--- a/pandas/tests/series/methods/test_interpolate.py
+++ b/pandas/tests/series/methods/test_interpolate.py
@@ -812,6 +812,23 @@ def test_interpolate_unsorted_index(self, ascending, expected_values):
         expected = Series(data=expected_values, index=expected_values, dtype=float)
         tm.assert_series_equal(result, expected)
 
+    def test_interpolate_duplicate_index_decreasing(self):
+        # GH 42585
+        ts = Series(data=[-200, -50, np.nan, -50, 1000], index=[79, 79, 0, -61, -2783])
+        result = ts.interpolate(method="index")
+        assert result.loc[0] == -50
+        # TODO: also check validity of sort assumption
+
+    def test_interpolate_duplicate_index_increasing(self):
+        # GH 42585
+        ts = Series(
+            data=[0, 0, 0, 0, 0, 0, 0, -50, np.nan, -50, 0, 0, 0, 0, 0, 0, 0, 0],
+            index=[1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11],
+        )
+        result = ts.interpolate(method="index")
+        assert result.loc[0] == -50
+        # TODO: check validity of sort assumption?
+
     def test_interpolate_pos_args_deprecation(self):
         # https://github.com/pandas-dev/pandas/issues/41485
         ser = Series([1, 2, 3])