diff --git a/pyomo/solvers/plugins/solvers/cuopt_direct.py b/pyomo/solvers/plugins/solvers/cuopt_direct.py
index 87f16b861c9..4ccdc9e631d 100644
--- a/pyomo/solvers/plugins/solvers/cuopt_direct.py
+++ b/pyomo/solvers/plugins/solvers/cuopt_direct.py
@@ -249,16 +249,18 @@ def _postsolve(self):
         is_mip = solution.get_problem_category()
 
         # Termination Status
-        # 0 - CUOPT_TERIMINATION_STATUS_NO_TERMINATION
-        # 1 - CUOPT_TERIMINATION_STATUS_OPTIMAL
-        # 2 - CUOPT_TERIMINATION_STATUS_INFEASIBLE
-        # 3 - CUOPT_TERIMINATION_STATUS_UNBOUNDED
-        # 4 - CUOPT_TERIMINATION_STATUS_ITERATION_LIMIT
-        # 5 - CUOPT_TERIMINATION_STATUS_TIME_LIMIT
-        # 6 - CUOPT_TERIMINATION_STATUS_NUMERICAL_ERROR
-        # 7 - CUOPT_TERIMINATION_STATUS_PRIMAL_FEASIBLE
-        # 8 - CUOPT_TERIMINATION_STATUS_FEASIBLE_FOUND
-        # 9 - CUOPT_TERIMINATION_STATUS_CONCURRENT_LIMIT
+        #  0 - CUOPT_TERIMINATION_STATUS_NO_TERMINATION
+        #  1 - CUOPT_TERIMINATION_STATUS_OPTIMAL
+        #  2 - CUOPT_TERIMINATION_STATUS_INFEASIBLE
+        #  3 - CUOPT_TERIMINATION_STATUS_UNBOUNDED
+        #  4 - CUOPT_TERIMINATION_STATUS_ITERATION_LIMIT
+        #  5 - CUOPT_TERIMINATION_STATUS_TIME_LIMIT
+        #  6 - CUOPT_TERIMINATION_STATUS_NUMERICAL_ERROR
+        #  7 - CUOPT_TERIMINATION_STATUS_PRIMAL_FEASIBLE
+        #  8 - CUOPT_TERIMINATION_STATUS_FEASIBLE_FOUND
+        #  9 - CUOPT_TERIMINATION_STATUS_CONCURRENT_LIMIT
+        # 10 - CUOPT_TERIMINATION_STATUS_WORK_LIMIT
+        # 11 - CUOPT_TERIMINATION_STATUS_UNBOUNDED_OR_INFEASIBLE
 
         if status == 1:
             self.results.solver.status = SolverStatus.ok
@@ -292,6 +294,12 @@ def _postsolve(self):
             self.results.solver.status = SolverStatus.ok
             self.results.solver.termination_condition = TerminationCondition.other
             soln.status = SolutionStatus.other
+        elif status == 11:
+            self.results.solver.status = SolverStatus.warning
+            self.results.solver.termination_condition = (
+                TerminationCondition.infeasibleOrUnbounded
+            )
+            soln.status = SolutionStatus.unsure
         else:
             self.results.solver.status = SolverStatus.error
             self.results.solver.termination_condition = TerminationCondition.error
diff --git a/pyomo/solvers/tests/checks/test_cuopt_direct.py b/pyomo/solvers/tests/checks/test_cuopt_direct.py
index aef0a924468..baa13c560e8 100644
--- a/pyomo/solvers/tests/checks/test_cuopt_direct.py
+++ b/pyomo/solvers/tests/checks/test_cuopt_direct.py
@@ -125,6 +125,23 @@ def test_infeasible_trivial_constraint(self):
         with pytest.raises(ValueError, match=r"Trivial constraint.*infeasible"):
             opt.solve(m, skip_trivial_constraints=True)
 
+    @unittest.skipIf(not cuopt_available, "The CuOpt solver is not available")
+    def test_unbounded_or_infeasible_status(self):
+        # An LP with no variable bounds and an unbounded objective triggers
+        # cuOpt's presolver to return UnboundedOrInfeasible (status 11), which
+        # the plugin maps to TerminationCondition.infeasibleOrUnbounded.
+        m = ConcreteModel()
+        m.x = Var()
+        m.y = Var()
+        m.obj = Objective(expr=m.x + m.y, sense=minimize)
+
+        opt = SolverFactory('cuopt')
+        res = opt.solve(m, load_solutions=False)
+
+        self.assertEqual(res.solver.termination_condition, "infeasibleOrUnbounded")
+        self.assertEqual(res.solver.status, "warning")
+        self.assertEqual(res.solution[0].status, "unsure")
+
     @unittest.skipIf(not cuopt_available, "The CuOpt solver is not available")
     def test_nonlinear_constraint_rejected(self):
         m = ConcreteModel()