diff --git a/crates/accelerate/src/consolidate_blocks.rs b/crates/accelerate/src/consolidate_blocks.rs
index 1edd592ce87..0fec3fa2909 100644
--- a/crates/accelerate/src/consolidate_blocks.rs
+++ b/crates/accelerate/src/consolidate_blocks.rs
@@ -54,11 +54,12 @@ const MAX_2Q_DEPTH: usize = 20;
 
 #[allow(clippy::too_many_arguments)]
 #[pyfunction]
-#[pyo3(signature = (dag, decomposer, force_consolidate, target=None, basis_gates=None, blocks=None, runs=None))]
+#[pyo3(signature = (dag, decomposer, basis_gate_name, force_consolidate, target=None, basis_gates=None, blocks=None, runs=None))]
 pub(crate) fn consolidate_blocks(
     py: Python,
     dag: &mut DAGCircuit,
     decomposer: &TwoQubitBasisDecomposer,
+    basis_gate_name: &str,
     force_consolidate: bool,
     target: Option<&Target>,
     basis_gates: Option<HashSet<String>>,
@@ -125,7 +126,7 @@ pub(crate) fn consolidate_blocks(
             let inst = dag.dag()[*node].unwrap_operation();
             block_qargs.extend(dag.get_qargs(inst.qubits));
             all_block_gates.insert(*node);
-            if inst.op.name() == decomposer.gate_name() {
+            if inst.op.name() == basis_gate_name {
                 basis_count += 1;
             }
             if !is_supported(
diff --git a/qiskit/transpiler/passes/optimization/consolidate_blocks.py b/qiskit/transpiler/passes/optimization/consolidate_blocks.py
index ce64233699b..f31401abb6a 100644
--- a/qiskit/transpiler/passes/optimization/consolidate_blocks.py
+++ b/qiskit/transpiler/passes/optimization/consolidate_blocks.py
@@ -109,6 +109,7 @@ def run(self, dag):
         consolidate_blocks(
             dag,
             self.decomposer._inner_decomposer,
+            self.decomposer.gate.name,
             self.force_consolidate,
             target=self.target,
             basis_gates=self.basis_gates,
diff --git a/test/python/transpiler/test_consolidate_blocks.py b/test/python/transpiler/test_consolidate_blocks.py
index fbe11ff77db..83379a4eecb 100644
--- a/test/python/transpiler/test_consolidate_blocks.py
+++ b/test/python/transpiler/test_consolidate_blocks.py
@@ -18,8 +18,17 @@
 import numpy as np
 from ddt import ddt, data
 
-from qiskit.circuit import QuantumCircuit, QuantumRegister, IfElseOp, Gate
-from qiskit.circuit.library import U2Gate, SwapGate, CXGate, CZGate, UnitaryGate
+from qiskit.circuit import QuantumCircuit, QuantumRegister, IfElseOp, Gate, Parameter
+from qiskit.circuit.library import (
+    U2Gate,
+    SwapGate,
+    CXGate,
+    CZGate,
+    UnitaryGate,
+    SXGate,
+    XGate,
+    RZGate,
+)
 from qiskit.converters import circuit_to_dag
 from qiskit.quantum_info.operators import Operator
 from qiskit.quantum_info.operators.measures import process_fidelity
@@ -590,6 +599,73 @@ def test_no_kak_gates_in_preset_pm(self, opt_level):
         tqc = pm.run(qc)
         self.assertEqual(ref_tqc, tqc)
 
+    def test_non_cx_basis_gate(self):
+        """Test a non-cx kak gate is consolidated correctly."""
+        qc = QuantumCircuit(2)
+        qc.cz(0, 1)
+        qc.x(0)
+        qc.h(1)
+        qc.z(1)
+        qc.t(1)
+        qc.h(0)
+        qc.t(0)
+        qc.cz(1, 0)
+        qc.sx(0)
+        qc.sx(1)
+        qc.cz(0, 1)
+        qc.sx(0)
+        qc.sx(1)
+        qc.cz(1, 0)
+        qc.x(0)
+        qc.h(1)
+        qc.z(1)
+        qc.t(1)
+        qc.h(0)
+        qc.t(0)
+        qc.cz(0, 1)
+
+        consolidate_pass = ConsolidateBlocks(basis_gates=["sx", "x", "rz", "cz"])
+        res = consolidate_pass(qc)
+        self.assertEqual({"unitary": 1}, res.count_ops())
+        self.assertEqual(Operator.from_circuit(qc), Operator(res.data[0].operation.params[0]))
+
+    def test_non_cx_target(self):
+        """Test a non-cx kak gate is consolidated correctly."""
+        qc = QuantumCircuit(2)
+        qc.cz(0, 1)
+        qc.x(0)
+        qc.h(1)
+        qc.z(1)
+        qc.t(1)
+        qc.h(0)
+        qc.t(0)
+        qc.cz(1, 0)
+        qc.sx(0)
+        qc.sx(1)
+        qc.cz(0, 1)
+        qc.sx(0)
+        qc.sx(1)
+        qc.cz(1, 0)
+        qc.x(0)
+        qc.h(1)
+        qc.z(1)
+        qc.t(1)
+        qc.h(0)
+        qc.t(0)
+        qc.cz(0, 1)
+
+        phi = Parameter("phi")
+        target = Target(num_qubits=2)
+        target.add_instruction(SXGate(), {(0,): None, (1,): None})
+        target.add_instruction(XGate(), {(0,): None, (1,): None})
+        target.add_instruction(RZGate(phi), {(0,): None, (1,): None})
+        target.add_instruction(CZGate(), {(0, 1): None, (1, 0): None})
+
+        consolidate_pass = ConsolidateBlocks(target=target)
+        res = consolidate_pass(qc)
+        self.assertEqual({"unitary": 1}, res.count_ops())
+        self.assertEqual(Operator.from_circuit(qc), Operator(res.data[0].operation.params[0]))
+
 
 if __name__ == "__main__":
     unittest.main()