diff --git a/pyomo/core/base/param.py b/pyomo/core/base/param.py
index 03d700140e8..3ef33b9ee45 100644
--- a/pyomo/core/base/param.py
+++ b/pyomo/core/base/param.py
@@ -162,16 +162,31 @@ def set_value(self, value, idx=NOTSET):
         # required to be mutable.
         #
         _comp = self.parent_component()
-        if type(value) in native_types:
+        if value.__class__ in native_types:
             # TODO: warn/error: check if this Param has units: assigning
             # a dimensionless value to a united param should be an error
             pass
         elif _comp._units is not None:
             _src_magnitude = expr_value(value)
-            _src_units = units.get_units(value)
-            value = units.convert_value(
-                num_value=_src_magnitude, from_units=_src_units, to_units=_comp._units
-            )
+            # Note: expr_value() could have just registered a new numeric type
+            if value.__class__ in native_types:
+                value = _src_magnitude
+            else:
+                _src_units = units.get_units(value)
+                value = units.convert_value(
+                    num_value=_src_magnitude,
+                    from_units=_src_units,
+                    to_units=_comp._units,
+                )
+        # FIXME: we should call value() here [to ensure types get
+        # registered], but doing so breaks non-numeric Params (which we
+        # allow).  The real fix will be to follow the precedent from
+        # GetItemExpression and have separate types based on which
+        # expression "system" the Param should participate in (numeric,
+        # logical, or structural).
+        #
+        # else:
+        #     value = expr_value(value)
 
         old_value, self._value = self._value, value
         try:
diff --git a/pyomo/core/base/var.py b/pyomo/core/base/var.py
index d03fd0b677f..f426c9c4f55 100644
--- a/pyomo/core/base/var.py
+++ b/pyomo/core/base/var.py
@@ -384,17 +384,22 @@ def set_value(self, val, skip_validation=False):
         #
         # Check if this Var has units: assigning dimensionless
         # values to a variable with units should be an error
-        if type(val) not in native_numeric_types:
-            if self.parent_component()._units is not None:
-                _src_magnitude = value(val)
+        if val.__class__ in native_numeric_types:
+            pass
+        elif self.parent_component()._units is not None:
+            _src_magnitude = value(val)
+            # Note: value() could have just registered a new numeric type
+            if val.__class__ in native_numeric_types:
+                val = _src_magnitude
+            else:
                 _src_units = units.get_units(val)
                 val = units.convert_value(
                     num_value=_src_magnitude,
                     from_units=_src_units,
                     to_units=self.parent_component()._units,
                 )
-            else:
-                val = value(val)
+        else:
+            val = value(val)
 
         if not skip_validation:
             if val not in self.domain:
diff --git a/pyomo/core/tests/unit/test_numvalue.py b/pyomo/core/tests/unit/test_numvalue.py
index eceab3a42d9..bd784d655e8 100644
--- a/pyomo/core/tests/unit/test_numvalue.py
+++ b/pyomo/core/tests/unit/test_numvalue.py
@@ -562,7 +562,8 @@ def test_numpy_basic_bool_registration(self):
     @unittest.skipUnless(numpy_available, "This test requires NumPy")
     def test_automatic_numpy_registration(self):
         cmd = (
-            'import pyomo; from pyomo.core.base import Var, Param; import numpy as np; '
+            'import pyomo; from pyomo.core.base import Var, Param; '
+            'from pyomo.core.base.units_container import units; import numpy as np; '
             'print(np.float64 in pyomo.common.numeric_types.native_numeric_types); '
             '%s; print(np.float64 in pyomo.common.numeric_types.native_numeric_types)'
         )
@@ -582,6 +583,11 @@ def _tester(expr):
         _tester('Var() + np.float64(5)')
         _tester('v = Var(); v.construct(); v.value = np.float64(5)')
         _tester('p = Param(mutable=True); p.construct(); p.value = np.float64(5)')
+        _tester('v = Var(units=units.m); v.construct(); v.value = np.float64(5)')
+        _tester(
+            'p = Param(mutable=True, units=units.m); p.construct(); '
+            'p.value = np.float64(5)'
+        )
 
 
 if __name__ == "__main__":