diff --git a/qiskit/transpiler/passes/synthesis/hls_plugins.py b/qiskit/transpiler/passes/synthesis/hls_plugins.py
index 3dce030c1982..ff483c9a5643 100644
--- a/qiskit/transpiler/passes/synthesis/hls_plugins.py
+++ b/qiskit/transpiler/passes/synthesis/hls_plugins.py
@@ -505,6 +505,10 @@ class QFTSynthesisFull(HighLevelSynthesisPlugin):
     This plugin name is :``qft.full`` which can be used as the key on
     an :class:`~.HLSConfig` object to use this method with :class:`~.HighLevelSynthesis`.
 
+    Note that the plugin mechanism is not applied if the gate is called ``qft`` but
+    is not an instance of ``QFTGate``. This allows users to create custom gates with
+    name ``qft``.
+
     The plugin supports the following additional options:
 
     * reverse_qubits (bool): Whether to synthesize the "QFT" operation (if ``False``,
@@ -532,10 +536,11 @@ class QFTSynthesisFull(HighLevelSynthesisPlugin):
 
     def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **options):
         """Run synthesis for the given QFTGate."""
+
+        # Even though the gate is called "qft", it's not a QFTGate,
+        # and we should not synthesize it using the plugin.
         if not isinstance(high_level_object, QFTGate):
-            raise TranspilerError(
-                "The synthesis plugin 'qft.full` only applies to objects of type QFTGate."
-            )
+            return None
 
         reverse_qubits = options.get("reverse_qubits", False)
         approximation_degree = options.get("approximation_degree", 0)
@@ -560,6 +565,10 @@ class QFTSynthesisLine(HighLevelSynthesisPlugin):
     This plugin name is :``qft.line`` which can be used as the key on
     an :class:`~.HLSConfig` object to use this method with :class:`~.HighLevelSynthesis`.
 
+    Note that the plugin mechanism is not applied if the gate is called ``qft`` but
+    is not an instance of ``QFTGate``. This allows users to create custom gates with
+    name ``qft``.
+
     The plugin supports the following additional options:
 
     * reverse_qubits (bool): Whether to synthesize the "QFT" operation (if ``False``,
@@ -584,10 +593,11 @@ class QFTSynthesisLine(HighLevelSynthesisPlugin):
 
     def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **options):
         """Run synthesis for the given QFTGate."""
+
+        # Even though the gate is called "qft", it's not a QFTGate,
+        # and we should not synthesize it using the plugin.
         if not isinstance(high_level_object, QFTGate):
-            raise TranspilerError(
-                "The synthesis plugin 'qft.line` only applies to objects of type QFTGate."
-            )
+            return None
 
         reverse_qubits = options.get("reverse_qubits", False)
         approximation_degree = options.get("approximation_degree", 0)
diff --git a/releasenotes/notes/fix-qft-plugins-7106029d33c44b96.yaml b/releasenotes/notes/fix-qft-plugins-7106029d33c44b96.yaml
new file mode 100644
index 000000000000..3024432af714
--- /dev/null
+++ b/releasenotes/notes/fix-qft-plugins-7106029d33c44b96.yaml
@@ -0,0 +1,9 @@
+---
+fixes:
+  - |
+    The ``HighLevelSynthesis`` transpiler pass no longer raises an exception when
+    encountering a custom gate that is called "qft" but is not an instance of
+    :class:`~qiskit.circuit.library.QFTGate`. Instead, the synthesis plugins for
+    QFT gates ignore such a gate, and the gate's definition is used (if provided).
+
+
diff --git a/test/python/transpiler/test_high_level_synthesis.py b/test/python/transpiler/test_high_level_synthesis.py
index f3b11c98bfc7..a7dd806e63e9 100644
--- a/test/python/transpiler/test_high_level_synthesis.py
+++ b/test/python/transpiler/test_high_level_synthesis.py
@@ -2309,6 +2309,22 @@ def test_qft_line_plugin_annotated_qft(self, qft_plugin_name):
         qct = hls_pass(qc)
         self.assertEqual(Operator(qc), Operator(qct))
 
+    @data("line", "full")
+    def test_skip_non_qft(self, qft_plugin_name):
+        """Test that synthesis plugins are not applied on gates that are called `qft`, yet
+        that are not of type `QFTGate`.
+        """
+        qc = QuantumCircuit(1)
+        qc2 = QuantumCircuit(1, name="qft")
+        qc2.s(0)
+        qc.append(qc2.to_instruction(), qc.qregs[0])
+        hls_config = HLSConfig(qft=[qft_plugin_name])
+        hls_pass = HighLevelSynthesis(hls_config=hls_config)
+        qct = hls_pass(qc)
+        # HighLevelSynthesis should replace the custom gate called "qft"
+        # by the user-provided definition.
+        self.assertEqual(Operator(qc2), Operator(qct))
+
 
 @ddt
 class TestMCXSynthesisPlugins(QiskitTestCase):