diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 5bdf6536d3d..ac80524a3c4 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -91,7 +91,8 @@ Bug fixes
   to preserve attributes. :py:meth:`Dataset.coarsen` accepts a keyword
   argument ``keep_attrs`` to change this setting. (:issue:`3376`,
   :pull:`3801`) By `Andrew Thomas <https://github.com/amcnicho>`_.
-  
+- Delete associated indexes when deleting coordinate variables. (:issue:`3746`).
+  By `Deepak Cherian <https://github.com/dcherian>`_.
 - Fix :py:meth:`xarray.core.dataset.Dataset.to_zarr` when using `append_dim` and `group`
   simultaneously. (:issue:`3170`). By `Matthias Meyer <https://github.com/niowniow>`_.
 - Fix html repr on :py:class:`Dataset` with non-string keys (:pull:`3807`).
diff --git a/xarray/core/coordinates.py b/xarray/core/coordinates.py
index 3d51c9b4271..83c4d2a8636 100644
--- a/xarray/core/coordinates.py
+++ b/xarray/core/coordinates.py
@@ -247,7 +247,7 @@ def __delitem__(self, key: Hashable) -> None:
         if key in self:
             del self._data[key]
         else:
-            raise KeyError(key)
+            raise KeyError(f"{key!r} is not a coordinate variable.")
 
     def _ipython_key_completions_(self):
         """Provide method for the key-autocompletions in IPython. """
@@ -291,7 +291,7 @@ def _update_coords(
         dims = calculate_dimensions(coords_plus_data)
         if not set(dims) <= set(self.dims):
             raise ValueError(
-                "cannot add coordinates with new dimensions to " "a DataArray"
+                "cannot add coordinates with new dimensions to a DataArray"
             )
         self._data._coords = coords
 
@@ -312,7 +312,12 @@ def to_dataset(self) -> "Dataset":
         return Dataset._construct_direct(coords, set(coords))
 
     def __delitem__(self, key: Hashable) -> None:
-        del self._data._coords[key]
+        if key in self:
+            del self._data._coords[key]
+            if self._data._indexes is not None and key in self._data._indexes:
+                del self._data._indexes[key]
+        else:
+            raise KeyError(f"{key!r} is not a coordinate variable.")
 
     def _ipython_key_completions_(self):
         """Provide method for the key-autocompletions in IPython. """
diff --git a/xarray/tests/test_dataarray.py b/xarray/tests/test_dataarray.py
index 6f065c9daed..fbd9810f285 100644
--- a/xarray/tests/test_dataarray.py
+++ b/xarray/tests/test_dataarray.py
@@ -1412,6 +1412,12 @@ def test_coords_non_string(self):
         expected = DataArray(2, coords={1: 2}, name=1)
         assert_identical(actual, expected)
 
+    def test_coords_delitem_delete_indexes(self):
+        # regression test for GH3746
+        arr = DataArray(np.ones((2,)), dims="x", coords={"x": [0, 1]})
+        del arr.coords["x"]
+        assert "x" not in arr.indexes
+
     def test_broadcast_like(self):
         arr1 = DataArray(
             np.ones((2, 3)),
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index d2e7bcdabf8..20b814a25c7 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -744,6 +744,10 @@ def test_coords_modify(self):
         expected = data.merge({"c": 11}).set_coords("c")
         assert_identical(expected, actual)
 
+        # regression test for GH3746
+        del actual.coords["x"]
+        assert "x" not in actual.indexes
+
     def test_update_index(self):
         actual = Dataset(coords={"x": [1, 2, 3]})
         actual["x"] = ["a", "b", "c"]