diff --git a/doc/source/whatsnew/v1.1.3.rst b/doc/source/whatsnew/v1.1.3.rst
index 25d223418fc92..5cbd160f29d66 100644
--- a/doc/source/whatsnew/v1.1.3.rst
+++ b/doc/source/whatsnew/v1.1.3.rst
@@ -25,6 +25,7 @@ Fixed regressions
 Bug fixes
 ~~~~~~~~~
 - Bug in :meth:`Series.str.startswith` and :meth:`Series.str.endswith` with ``category`` dtype not propagating ``na`` parameter (:issue:`36241`)
+- Bug in :class:`Series` constructor where integer overflow would occur for sufficiently large scalar inputs when an index was provided (:issue:`36291`)
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/core/dtypes/cast.py b/pandas/core/dtypes/cast.py
index 64ccc0be0a25d..05759ffb43dde 100644
--- a/pandas/core/dtypes/cast.py
+++ b/pandas/core/dtypes/cast.py
@@ -697,6 +697,11 @@ def infer_dtype_from_scalar(val, pandas_dtype: bool = False) -> Tuple[DtypeObj,
         else:
             dtype = np.dtype(np.int64)
 
+        try:
+            np.array(val, dtype=dtype)
+        except OverflowError:
+            dtype = np.array(val).dtype
+
     elif is_float(val):
         if isinstance(val, np.floating):
             dtype = np.dtype(type(val))
diff --git a/pandas/tests/series/test_constructors.py b/pandas/tests/series/test_constructors.py
index 0fb8c5955a2e7..8ac0a55e63cd1 100644
--- a/pandas/tests/series/test_constructors.py
+++ b/pandas/tests/series/test_constructors.py
@@ -1501,3 +1501,10 @@ def test_construction_from_ordered_collection(self):
         result = Series({"a": 1, "b": 2}.values())
         expected = Series([1, 2])
         tm.assert_series_equal(result, expected)
+
+    def test_construction_from_large_int_scalar_no_overflow(self):
+        # https://github.com/pandas-dev/pandas/issues/36291
+        n = 1_000_000_000_000_000_000_000
+        result = Series(n, index=[0])
+        expected = Series(n)
+        tm.assert_series_equal(result, expected)