diff --git a/include/pybind11/cast.h b/include/pybind11/cast.h
index 6079283ffe..64683f8e4b 100644
--- a/include/pybind11/cast.h
+++ b/include/pybind11/cast.h
@@ -1040,26 +1040,30 @@ struct type_caster<T, enable_if_t<std::is_arithmetic<T>::value && !is_std_char_t
                 return false;
         } else if (PyFloat_Check(src.ptr())) {
             return false;
-        } else if (!convert && !index_check(src.ptr()) && !PYBIND11_LONG_CHECK(src.ptr())) {
+        } else if (!convert && !PYBIND11_LONG_CHECK(src.ptr()) && !index_check(src.ptr())) {
             return false;
         } else {
             handle src_or_index = src;
-            py_value = (py_type) -1;
 #if PY_VERSION_HEX < 0x03080000
             object index;
-            if (!PYBIND11_LONG_CHECK(src.ptr()) && index_check(src.ptr())) {
+            if (!PYBIND11_LONG_CHECK(src.ptr())) {  // So: index_check(src.ptr())
                 index = reinterpret_steal<object>(PyNumber_Index(src.ptr()));
-                src_or_index = index ? index : handle();
+                if (!index) {
+                    PyErr_Clear();
+                    if (!convert)
+                        return false;
+                }
+                else {
+                    src_or_index = index;
+                }
             }
 #endif
-            if (src_or_index) {
-                if (std::is_unsigned<py_type>::value) {
-                    py_value = as_unsigned<py_type>(src_or_index.ptr());
-                } else { // signed integer:
-                    py_value = sizeof(T) <= sizeof(long)
-                        ? (py_type) PyLong_AsLong(src_or_index.ptr())
-                        : (py_type) PYBIND11_LONG_AS_LONGLONG(src_or_index.ptr());
-                }
+            if (std::is_unsigned<py_type>::value) {
+                py_value = as_unsigned<py_type>(src_or_index.ptr());
+            } else { // signed integer:
+                py_value = sizeof(T) <= sizeof(long)
+                    ? (py_type) PyLong_AsLong(src_or_index.ptr())
+                    : (py_type) PYBIND11_LONG_AS_LONGLONG(src_or_index.ptr());
             }
         }
 
@@ -1069,15 +1073,8 @@ struct type_caster<T, enable_if_t<std::is_arithmetic<T>::value && !is_std_char_t
         // Check to see if the conversion is valid (integers should match exactly)
         // Signed/unsigned checks happen elsewhere
         if (py_err || (std::is_integral<T>::value && sizeof(py_type) != sizeof(T) && py_value != (py_type) (T) py_value)) {
-            bool type_error = py_err && PyErr_ExceptionMatches(
-#if PY_VERSION_HEX < 0x03000000 && !defined(PYPY_VERSION)
-                PyExc_SystemError
-#else
-                PyExc_TypeError
-#endif
-            );
             PyErr_Clear();
-            if (type_error && convert && PyNumber_Check(src.ptr())) {
+            if (py_err && convert && PyNumber_Check(src.ptr())) {
                 auto tmp = reinterpret_steal<object>(std::is_floating_point<T>::value
                                                      ? PyNumber_Float(src.ptr())
                                                      : PyNumber_Long(src.ptr()));
diff --git a/tests/test_builtin_casters.py b/tests/test_builtin_casters.py
index cfa07f04b5..6385346014 100644
--- a/tests/test_builtin_casters.py
+++ b/tests/test_builtin_casters.py
@@ -251,12 +251,18 @@ def test_integer_casting():
         assert "incompatible function arguments" in str(excinfo.value)
 
 
-@pytest.mark.filterwarnings("ignore:an integer is required:DeprecationWarning")
 def test_int_convert():
     class DeepThought(object):
         def __int__(self):
             return 42
 
+    class DoubleThought(object):
+        def __int__(self):
+            return 42
+
+        def __index__(self):
+            return 0
+
     class ShallowThought(object):
         pass
 
@@ -284,7 +290,7 @@ def __int__(self):
 
     convert, noconvert = m.int_passthrough, m.int_passthrough_noconvert
 
-    def require_implicit(v):
+    def requires_conversion(v):
         pytest.raises(TypeError, noconvert, v)
 
     def cant_convert(v):
@@ -294,20 +300,22 @@ def cant_convert(v):
     assert noconvert(7) == 7
     cant_convert(3.14159)
     assert convert(DeepThought()) == 42
-    require_implicit(DeepThought())
+    requires_conversion(DeepThought())
+    assert convert(DoubleThought()) == 0  # Fishy; `int(DoubleThought)` == 42
+    assert noconvert(DoubleThought()) == 0
     cant_convert(ShallowThought())
     cant_convert(FuzzyThought())
 
-    # Before Python 3.8, `int(obj)` does not pick up on `obj.__index__`, but pybind11
-    # "backports" this behavior.
+    # Before Python 3.8, `PyLong_AsLong` does not pick up on `obj.__index__`,
+    # but pybind11 "backports" this behavior.
     assert convert(IndexedThought()) == 42
     assert noconvert(IndexedThought()) == 42
     assert convert(TypeErrorThought()) == 42
-    require_implicit(TypeErrorThought())
-    cant_convert(RaisingThought())  # no fall-back to `__int__`if `__index__` raises
+    requires_conversion(TypeErrorThought())
+    assert convert(RaisingThought()) == 42
+    requires_conversion(RaisingThought())
 
 
-@pytest.mark.filterwarnings("ignore:an integer is required:DeprecationWarning")
 def test_numpy_int_convert():
     np = pytest.importorskip("numpy")