diff --git a/qiskit/transpiler/preset_passmanagers/builtin_plugins.py b/qiskit/transpiler/preset_passmanagers/builtin_plugins.py
index 7e3e2611546f..3ef95eedd5e1 100644
--- a/qiskit/transpiler/preset_passmanagers/builtin_plugins.py
+++ b/qiskit/transpiler/preset_passmanagers/builtin_plugins.py
@@ -150,7 +150,8 @@ def pass_manager(self, pass_manager_config, optimization_level=None) -> PassMana
                 pass_manager_config.unitary_synthesis_plugin_config,
                 pass_manager_config.hls_config,
             )
-            init.append(ElidePermutations())
+            if pass_manager_config.routing_method != "none":
+                init.append(ElidePermutations())
             init.append(RemoveDiagonalGatesBeforeMeasure())
             init.append(
                 InverseCancellation(
diff --git a/releasenotes/notes/no-elide-routing-none-7c1bebf1283d48c0.yaml b/releasenotes/notes/no-elide-routing-none-7c1bebf1283d48c0.yaml
new file mode 100644
index 000000000000..cc749f62775c
--- /dev/null
+++ b/releasenotes/notes/no-elide-routing-none-7c1bebf1283d48c0.yaml
@@ -0,0 +1,16 @@
+---
+fixes:
+  - |
+    Fixed an issue when running :func:`.transpile` or
+    :meth:`~.StagedPassManager.run` on a pass manager generated by
+    :func:`.generate_preset_pass_manager` using ``optimization_level`` 2 or 3
+    when the ``routing_method`` argument is set to ``"none"`` to explicitly
+    disable routing. Previously under these conditions the transpiler would run
+    the :class:`.ElidePermutation` pass as part of the init stage as under normal
+    conditions this is a useful optimization to remove :class:`.SwapGate`
+    and :class:`.PermutationGate` instances from the circuit. But when
+    ``routing_method="none"`` this optimization wasn't expected as it permutes
+    the circuit in a similar manner to routing which shouldn't be performed
+    when ``routing_method="none"``. This has been fixed by no longer
+    running :class:`.ElidePermutation` if ``routing_method="none"`` is set.
+    Fixed `#13144 <https://github.com/Qiskit/qiskit/issues/13144>`__
diff --git a/test/python/compiler/test_transpiler.py b/test/python/compiler/test_transpiler.py
index a29bfc21c8bb..5307d07428f1 100644
--- a/test/python/compiler/test_transpiler.py
+++ b/test/python/compiler/test_transpiler.py
@@ -863,6 +863,33 @@ def test_do_not_run_gatedirection_with_symmetric_cm(self):
             transpile(circ, coupling_map=coupling_map, initial_layout=layout)
             self.assertFalse(mock_pass.called)
 
+    def test_do_not_run_elide_permutations_no_routing(self):
+        """Test the ElidePermutations pass doesn't run if we disable routing
+
+        See https://github.com/Qiskit/qiskit/issues/13144 for the details and
+        reproduce in this test
+        """
+        circuit_routed = QuantumCircuit(4)
+        circuit_routed.cx(0, 1)
+        circuit_routed.h(1)
+        circuit_routed.swap(1, 2)
+        circuit_routed.cx(2, 3)
+        pm = generate_preset_pass_manager(
+            basis_gates=["cx", "sx", "rz"], routing_method="none", optimization_level=2
+        )
+        circuit_basis = pm.run(circuit_routed)
+        cx_gate_qubits = []
+        for instruction in circuit_basis.data:
+            if instruction.name == "cx":
+                cx_gate_qubits.append(instruction.qubits)
+        # If we did not Elide the existing swaps then the swap should be
+        # decomposed into 3 cx between 1 and 2 and there are no gates between
+        # 1 and 3
+        self.assertIn((circuit_basis.qubits[1], circuit_basis.qubits[2]), cx_gate_qubits)
+        self.assertIn((circuit_basis.qubits[2], circuit_basis.qubits[1]), cx_gate_qubits)
+        self.assertNotIn((circuit_basis.qubits[1], circuit_basis.qubits[3]), cx_gate_qubits)
+        self.assertNotIn((circuit_basis.qubits[3], circuit_basis.qubits[1]), cx_gate_qubits)
+
     def test_optimize_to_nothing(self):
         """Optimize gates up to fixed point in the default pipeline
         See https://github.com/Qiskit/qiskit-terra/issues/2035