diff --git a/pytensor/link/jax/linker.py b/pytensor/link/jax/linker.py
index 667806a80f..7a33910835 100644
--- a/pytensor/link/jax/linker.py
+++ b/pytensor/link/jax/linker.py
@@ -3,15 +3,19 @@
 from numpy.random import Generator, RandomState
 
 from pytensor.compile.sharedvalue import SharedVariable, shared
-from pytensor.graph.basic import Constant
 from pytensor.link.basic import JITLinker
 
 
 class JAXLinker(JITLinker):
     """A `Linker` that JIT-compiles NumPy-based operations using JAX."""
 
+    def __init__(self, *args, **kwargs):
+        self.scalar_shape_inputs: tuple[int] = ()  # type: ignore[annotation-unchecked]
+        super().__init__(*args, **kwargs)
+
     def fgraph_convert(self, fgraph, input_storage, storage_map, **kwargs):
         from pytensor.link.jax.dispatch import jax_funcify
+        from pytensor.link.jax.dispatch.shape import JAXShapeTuple
         from pytensor.tensor.random.type import RandomType
 
         shared_rng_inputs = [
@@ -63,6 +67,21 @@ def fgraph_convert(self, fgraph, input_storage, storage_map, **kwargs):
                 fgraph.inputs.remove(new_inp)
                 fgraph.inputs.insert(old_inp_fgrap_index, new_inp)
 
+        fgraph_inputs = fgraph.inputs
+        clients = fgraph.clients
+        # Detect scalar shape inputs that are used only in JAXShapeTuple nodes
+        scalar_shape_inputs = [
+            inp
+            for node in fgraph.apply_nodes
+            if isinstance(node.op, JAXShapeTuple)
+            for inp in node.inputs
+            if inp in fgraph_inputs
+            and all(isinstance(node.op, JAXShapeTuple) for node, _ in clients[inp])
+        ]
+        self.scalar_shape_inputs = tuple(
+            fgraph_inputs.index(inp) for inp in scalar_shape_inputs
+        )
+
         return jax_funcify(
             fgraph, input_storage=input_storage, storage_map=storage_map, **kwargs
         )
@@ -70,12 +89,19 @@ def fgraph_convert(self, fgraph, input_storage, storage_map, **kwargs):
     def jit_compile(self, fn):
         import jax
 
-        # I suppose we can consider `Constant`s to be "static" according to
-        # JAX.
-        static_argnums = [
-            n for n, i in enumerate(self.fgraph.inputs) if isinstance(i, Constant)
-        ]
-        return jax.jit(fn, static_argnums=static_argnums)
+        jit_fn = jax.jit(fn, static_argnums=self.scalar_shape_inputs)
+
+        if not self.scalar_shape_inputs:
+            return jit_fn
+
+        def convert_scalar_shape_inputs(*args):
+            new_args = [
+                int(arg) if i in self.scalar_shape_inputs else arg
+                for i, arg in enumerate(args)
+            ]
+            return jit_fn(*new_args)
+
+        return convert_scalar_shape_inputs
 
     def create_thunk_inputs(self, storage_map):
         from pytensor.link.jax.dispatch import jax_typify
diff --git a/tests/link/jax/test_random.py b/tests/link/jax/test_random.py
index 917c7be763..137b3241f0 100644
--- a/tests/link/jax/test_random.py
+++ b/tests/link/jax/test_random.py
@@ -863,15 +863,55 @@ def test_random_concrete_shape_subtensor_tuple(self):
         jax_fn = compile_random_function([x_pt], out)
         assert jax_fn(np.ones((2, 3))).shape == (2,)
 
+    def test_random_scalar_shape_input(self):
+        dim0 = pt.scalar("dim0", dtype=int)
+        dim1 = pt.scalar("dim1", dtype=int)
+
+        out = pt.random.normal(0, 1, size=dim0)
+        jax_fn = compile_random_function([dim0], out)
+        assert jax_fn(np.array(2)).shape == (2,)
+        assert jax_fn(np.array(3)).shape == (3,)
+
+        out = pt.random.normal(0, 1, size=[dim0, dim1])
+        jax_fn = compile_random_function([dim0, dim1], out)
+        assert jax_fn(np.array(2), np.array(3)).shape == (2, 3)
+        assert jax_fn(np.array(4), np.array(5)).shape == (4, 5)
+
     @pytest.mark.xfail(
-        reason="`size_pt` should be specified as a static argument", strict=True
+        raises=TypeError, reason="Cannot convert scalar input to integer"
     )
-    def test_random_concrete_shape_graph_input(self):
-        rng = shared(np.random.default_rng(123))
-        size_pt = pt.scalar()
-        out = pt.random.normal(0, 1, size=size_pt, rng=rng)
-        jax_fn = compile_random_function([size_pt], out)
-        assert jax_fn(10).shape == (10,)
+    def test_random_scalar_shape_input_not_supported(self):
+        dim = pt.scalar("dim", dtype=int)
+        out1 = pt.random.normal(0, 1, size=dim)
+        # An operation that wouldn't work if we replaced 0d array by integer
+        out2 = dim[...].set(1)
+        jax_fn = compile_random_function([dim], [out1, out2])
+
+        res1, res2 = jax_fn(np.array(2))
+        assert res1.shape == (2,)
+        assert res2 == 1
+
+    @pytest.mark.xfail(
+        raises=TypeError, reason="Cannot convert scalar input to integer"
+    )
+    def test_random_scalar_shape_input_not_supported2(self):
+        dim = pt.scalar("dim", dtype=int)
+        # This could theoretically be supported
+        # but would require knowing that * 2 is a safe operation for a python integer
+        out = pt.random.normal(0, 1, size=dim * 2)
+        jax_fn = compile_random_function([dim], out)
+        assert jax_fn(np.array(2)).shape == (4,)
+
+    @pytest.mark.xfail(
+        raises=TypeError, reason="Cannot convert tensor input to shape tuple"
+    )
+    def test_random_vector_shape_graph_input(self):
+        shape = pt.vector("shape", shape=(2,), dtype=int)
+        out = pt.random.normal(0, 1, size=shape)
+
+        jax_fn = compile_random_function([shape], out)
+        assert jax_fn(np.array([2, 3])).shape == (2, 3)
+        assert jax_fn(np.array([4, 5])).shape == (4, 5)
 
     def test_constant_shape_after_graph_rewriting(self):
         size = pt.vector("size", shape=(2,), dtype=int)