diff --git a/doc/source/whatsnew/v0.23.5.txt b/doc/source/whatsnew/v0.23.5.txt
index 916a246355b5f..f298dac8ae6a7 100644
--- a/doc/source/whatsnew/v0.23.5.txt
+++ b/doc/source/whatsnew/v0.23.5.txt
@@ -40,3 +40,7 @@ Bug Fixes
 
 -
 -
+
+**I/O**
+
+- Bug in :func:`read_csv` that caused it to raise ``OverflowError`` when trying to use 'inf' as ``na_value`` with integer index column (:issue:`17128`)
diff --git a/pandas/core/algorithms.py b/pandas/core/algorithms.py
index 49705cb6d9ad2..4bf62b021cddc 100644
--- a/pandas/core/algorithms.py
+++ b/pandas/core/algorithms.py
@@ -95,7 +95,7 @@ def _ensure_data(values, dtype=None):
                 values = ensure_float64(values)
             return values, 'float64', 'float64'
 
-    except (TypeError, ValueError):
+    except (TypeError, ValueError, OverflowError):
         # if we are trying to coerce to a dtype
         # and it is incompat this will fall thru to here
         return ensure_object(values), 'object', 'object'
@@ -429,7 +429,7 @@ def isin(comps, values):
             values = values.astype('int64', copy=False)
             comps = comps.astype('int64', copy=False)
             f = lambda x, y: htable.ismember_int64(x, y)
-        except (TypeError, ValueError):
+        except (TypeError, ValueError, OverflowError):
             values = values.astype(object)
             comps = comps.astype(object)
 
diff --git a/pandas/tests/io/parser/na_values.py b/pandas/tests/io/parser/na_values.py
index 9c68770e06b02..880ab707cfd07 100644
--- a/pandas/tests/io/parser/na_values.py
+++ b/pandas/tests/io/parser/na_values.py
@@ -369,3 +369,14 @@ def test_no_na_filter_on_index(self):
         expected = DataFrame({"a": [1, 4], "c": [3, 6]},
                              index=Index([np.nan, 5.0], name="b"))
         tm.assert_frame_equal(out, expected)
+
+    def test_inf_na_values_with_int_index(self):
+        # see gh-17128
+        data = "idx,col1,col2\n1,3,4\n2,inf,-inf"
+
+        # Don't fail with OverflowError with infs and integer index column
+        out = self.read_csv(StringIO(data), index_col=[0],
+                            na_values=['inf', '-inf'])
+        expected = DataFrame({"col1": [3, np.nan], "col2": [4, np.nan]},
+                             index=Index([1, 2], name="idx"))
+        tm.assert_frame_equal(out, expected)