diff --git a/test/onnx/test_pytorch_onnx_no_runtime.py b/test/onnx/test_pytorch_onnx_no_runtime.py
index 5f2ce3fa657a1ee..5f9d9980799df1c 100644
--- a/test/onnx/test_pytorch_onnx_no_runtime.py
+++ b/test/onnx/test_pytorch_onnx_no_runtime.py
@@ -777,6 +777,91 @@ def forward(self, x):
             model, inputs, f, dynamic_axes={"x": [0, 1]}, input_names=["x"]
         )
 
+    @common_utils.skipIfNoCaffe2
+    def test_caffe2_aten_fallback(self):
+        class ModelWithAtenNotONNXOp(torch.nn.Module):
+            def forward(self, x, y):
+                abcd = x + y
+                defg = torch.linalg.qr(abcd)
+                return defg
+
+        x = torch.rand(3, 4)
+        y = torch.rand(3, 4)
+        f = io.BytesIO()
+        torch.onnx.export(
+            ModelWithAtenNotONNXOp(),
+            (x, y),
+            f,
+            do_constant_folding=False,
+            operator_export_type=torch.onnx.OperatorExportTypes.ONNX_ATEN_FALLBACK,
+            # support for linalg.qr was added in later op set versions.
+            opset_version=9,
+        )
+        onnx_model = onnx.load(io.BytesIO(f.getvalue()))
+        self.assertAtenOp(onnx_model, "linalg_qr")
+
+    @common_utils.skipIfNoCaffe2
+    def test_caffe2_onnx_aten(self):
+        class ModelWithAtenFmod(torch.nn.Module):
+            def forward(self, x, y):
+                return torch.fmod(x, y)
+
+        x = torch.randn(3, 4, dtype=torch.float32)
+        y = torch.randn(3, 4, dtype=torch.float32)
+        f = io.BytesIO()
+        torch.onnx.export(
+            ModelWithAtenFmod(),
+            (x, y),
+            f,
+            do_constant_folding=False,
+            operator_export_type=torch.onnx.OperatorExportTypes.ONNX_ATEN,
+            opset_version=10,  # or higher
+        )
+        onnx_model = onnx.load(io.BytesIO(f.getvalue()))
+        assert onnx_model.graph.node[0].op_type == "Mod"
+
+    @common_utils.skipIfCaffe2
+    def test_aten_fallback(self):
+        class ModelWithAtenNotONNXOp(torch.nn.Module):
+            def forward(self, x, y):
+                abcd = x + y
+                defg = torch.linalg.qr(abcd)
+                return defg
+
+        x = torch.rand(3, 4)
+        y = torch.rand(3, 4)
+        f = io.BytesIO()
+        torch.onnx.export(
+            ModelWithAtenNotONNXOp(),
+            (x, y),
+            f,
+            do_constant_folding=False,
+            operator_export_type=torch.onnx.OperatorExportTypes.ONNX_ATEN_FALLBACK,
+            # support for linalg.qr was added in later op set versions.
+            opset_version=9,
+        )
+        onnx_model = onnx.load(io.BytesIO(f.getvalue()))
+        self.assertAtenOp(onnx_model, "linalg_qr")
+
+    @common_utils.skipIfCaffe2
+    def test_onnx_aten(self):
+        class ModelWithAtenFmod(torch.nn.Module):
+            def forward(self, x, y):
+                return torch.fmod(x, y)
+
+        x = torch.randn(3, 4, dtype=torch.float32)
+        y = torch.randn(3, 4, dtype=torch.float32)
+        f = io.BytesIO()
+        torch.onnx.export(
+            ModelWithAtenFmod(),
+            (x, y),
+            f,
+            do_constant_folding=False,
+            operator_export_type=torch.onnx.OperatorExportTypes.ONNX_ATEN,
+        )
+        onnx_model = onnx.load(io.BytesIO(f.getvalue()))
+        self.assertAtenOp(onnx_model, "fmod", "Tensor")
+
 
 if __name__ == "__main__":
     common_utils.run_tests()
diff --git a/torch/onnx/utils.py b/torch/onnx/utils.py
index 04fc984ded2b921..5e11c69baf0a7ff 100644
--- a/torch/onnx/utils.py
+++ b/torch/onnx/utils.py
@@ -1748,8 +1748,8 @@ def _should_aten_fallback(
     is_aten_fallback_export = (
         operator_export_type == _C_onnx.OperatorExportTypes.ONNX_ATEN_FALLBACK
     )
-    return is_onnx_aten_export or (
-        not is_exportable_aten_op and is_aten_fallback_export
+    return name.startswith("aten::") and (
+        is_onnx_aten_export or (not is_exportable_aten_op and is_aten_fallback_export)
     )
 
 
@@ -1844,6 +1844,30 @@ def _run_symbolic_function(
         env=env,
     )
 
+    # Direct ATen export requested
+    # For BUILD_CAFFE2=0 builds, if domain=="aten" and operator_export_type==ONNX_ATEN,
+    #   an aten::ATen operator is created regardless of symbolics existence
+    # For BUILD_CAFFE2=1, the same applies only if there is no symbolic available
+    if (
+        _C_onnx._CAFFE2_ATEN_FALLBACK
+        and not registration.registry.is_registered_op(ns_op_name, opset_version)
+    ) or (
+        not _C_onnx._CAFFE2_ATEN_FALLBACK
+        and _should_aten_fallback(ns_op_name, opset_version, operator_export_type)
+    ):
+        attrs = {
+            k + "_" + node.kindOf(k)[0]: symbolic_helper._node_get(node, k)
+            for k in node.attributeNames()
+        }
+        outputs = node.outputsSize()
+        attrs["outputs"] = outputs
+        return graph_context.at(
+            op_name,
+            *inputs,
+            overload_name=_get_aten_op_overload_name(node),
+            **attrs,
+        )
+
     try:
         # Caffe2-specific: Quantized op symbolics are registered for opset 9 only.
         if symbolic_helper.is_caffe2_aten_fallback() and opset_version == 9:
@@ -1861,6 +1885,7 @@ def _run_symbolic_function(
         if symbolic_function_group is not None:
             symbolic_fn = symbolic_function_group.get(opset_version)
             if symbolic_fn is not None:
+                # TODO Wrap almost identical attrs assignment or comment the difference.
                 attrs = {
                     k: symbolic_helper._node_get(node, k) for k in node.attributeNames()
                 }
@@ -1874,18 +1899,6 @@ def _run_symbolic_function(
             # Clone node to trigger ONNX shape inference
             return graph_context.op(op_name, *inputs, **attrs, outputs=node.outputsSize())  # type: ignore[attr-defined]
 
-        if _should_aten_fallback(ns_op_name, opset_version, operator_export_type):
-            # Direct ATen export requested
-            outputs = node.outputsSize()
-            attrs["outputs"] = outputs
-            # `overload_name` is set for non-Caffe2 builds only
-            return graph_context.at(
-                op_name,
-                *inputs,
-                overload_name=_get_aten_op_overload_name(node),
-                **attrs,
-            )
-
         raise errors.UnsupportedOperatorError(
             domain,
             op_name,