diff --git a/tensorflow/python/eager/function.py b/tensorflow/python/eager/function.py
index d1127fd0f45..199671ac6a0 100644
--- a/tensorflow/python/eager/function.py
+++ b/tensorflow/python/eager/function.py
@@ -84,47 +84,80 @@ def _flat_shape_list(*params):
           for x in nest.flatten(params)]
 
 
-def _compatible_shapes(flat_x, flat_y):
-  """Check if lists of TensorShapes contain compatible shapes.
+def _shape_less_specific_than(relaxed, to_check):
+  """Checks if `relaxed` is less specific than `to_check`.
+
+  This is an asymmetric check, unlike `TensorShape.is_compatible_with`. If
+  `to_check` has a dimension with an undefined shape, `relaxed` must also have
+  an undefined shape for that dimension.
 
   Args:
-    flat_x: List of TensorShape or None.
-    flat_y: List of TensorShape or None.
+    relaxed: A `TensorShape` to check against.
+    to_check: A second `TensorShape`.
+
+  Returns:
+    True if `to_check` represents a set of shapes which is a subset of
+    `relaxed`'s shapes and False otherwise.
+  """
+  if to_check.dims is not None and relaxed.dims is not None:
+    if to_check.rank != relaxed.rank:
+      return False
+    for check_dim, relaxed_dim in zip(to_check.dims, relaxed.dims):
+      if check_dim.value is None and relaxed_dim.value is not None:
+        return False
+      if not relaxed_dim.is_compatible_with(check_dim):
+        return False
+  return True
+
+
+def _compatible_shapes(flat_relaxed, flat_to_check):
+  """Check if lists of TensorShapes contain compatible shapes.
+
+  Checks that each `flat_relaxed` shape covers a superset of the shapes of the
+  corresponding `flat_to_check` shape.
+
+  Args:
+    flat_relaxed: List of TensorShape or None.
+    flat_to_check: List of TensorShape or None.
 
   Returns:
     A python bool.
 
   Raises:
-    RuntimeError: if `len(flat_x) != len(flat_y)`.
-    RuntimeError: if `flat_x[i] is None != flat_y[i] is None` for any `i`.
+    RuntimeError:
+      if `len(flat_relaxed) != len(flat_to_check)`.
+    RuntimeError:
+      if `flat_relaxed[i] is None != flat_to_check[i] is None` for any `i`.
   """
-  if len(flat_x) != len(flat_y):
+
+  if len(flat_relaxed) != len(flat_to_check):
     raise RuntimeError("Expected shape lists of identical lengths, but saw: "
-                       "%s and %s" % (flat_x, flat_y))
-  def is_compatible(x, y):
+                       "%s and %s" % (flat_relaxed, flat_to_check))
+  def is_compatible(relaxed, to_check):
     """Internal help function.
 
     Args:
-      x: TensorShape or None.
-      y: TensorShape or None.
+      relaxed: TensorShape or None.
+      to_check: TensorShape or None.
 
     Returns:
       Python bool.
 
     Raises:
-      RuntimeError: If `x is None != y is None`.
+      RuntimeError: If `relaxed is None != to_check is None`.
     """
     # If both x and y are None, there is no shape to compare.  Otherwise check
     # if they are compatible with each other.  Either way, both input signatures
     # must have have Tensors in the same entries.  If not, raise an assertion
     # error.
-    if x is None != y is None:
+    if relaxed is None != to_check is None:
       raise RuntimeError(
           "Expected signature type matches between flattened input shapes "
           "%s and %s; but saw that (%s is None) != (%s is None)"
-          % (flat_x, flat_y, x, y))
-    return x is None or x.is_compatible_with(y)
-  return all(is_compatible(x, y) for x, y in zip(flat_x, flat_y))
+          % (flat_relaxed, flat_to_check, relaxed, to_check))
+    return relaxed is None or _shape_less_specific_than(relaxed, to_check)
+  return all(is_compatible(relaxed, to_check)
+             for relaxed, to_check in zip(flat_relaxed, flat_to_check))
 
 
 def _common_shape(x, y):
@@ -1558,7 +1591,8 @@ class Function(object):
           rank_only_cache_key, None)
 
       if (relaxed_arg_function is not None
-          and _compatible_shapes(relaxed_arg_shapes, arg_shapes)):
+          and _compatible_shapes(flat_relaxed=relaxed_arg_shapes,
+                                 flat_to_check=arg_shapes)):
         return relaxed_arg_function, args, kwargs
 
       if relaxed_arg_shapes is None:
diff --git a/tensorflow/python/eager/function_test.py b/tensorflow/python/eager/function_test.py
index d86c1001e6c..3b80064db20 100644
--- a/tensorflow/python/eager/function_test.py
+++ b/tensorflow/python/eager/function_test.py
@@ -2535,6 +2535,24 @@ class FunctionTest(test.TestCase, parameterized.TestCase):
       x = func()
       self.assertRegexpMatches(x.device, 'GPU')
 
+  @test_util.run_in_graph_and_eager_modes
+  def testShapeCaching(self):
+
+    @function.defun
+    def func(x):
+      return array_ops.shape(x)
+
+    @function.defun(
+        input_signature=[tensor_spec.TensorSpec([None, None], dtypes.float32)])
+    def calls_func(x):
+      return func(x)
+
+    self.assertAllEqual([1, 1], self.evaluate(func(array_ops.zeros([1, 1]))))
+    self.assertAllEqual([2, 2], self.evaluate(func(array_ops.zeros([2, 2]))))
+    self.assertAllEqual(
+        [3, 3],
+        self.evaluate(calls_func(array_ops.zeros([3, 3]))))
+
 
 class MultiDeviceTest(test.TestCase, parameterized.TestCase):