diff --git a/doc/source/whatsnew/v1.3.4.rst b/doc/source/whatsnew/v1.3.4.rst
index 9387483bd62f5..4bbd44b395a35 100644
--- a/doc/source/whatsnew/v1.3.4.rst
+++ b/doc/source/whatsnew/v1.3.4.rst
@@ -16,6 +16,7 @@ Fixed regressions
 ~~~~~~~~~~~~~~~~~
 - Fixed regression in :meth:`merge` with integer and ``NaN`` keys failing with ``outer`` merge (:issue:`43550`)
 - Fixed performance regression in :meth:`MultiIndex.equals` (:issue:`43549`)
+- Fixed regression in :meth:`DataFrame.corr` raising ``ValueError`` with ``method="spearman`` on 32-bit platforms (:issue:`43588`)
 -
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/_libs/algos.pyx b/pandas/_libs/algos.pyx
index f619f09f85e66..d574df772d7e2 100644
--- a/pandas/_libs/algos.pyx
+++ b/pandas/_libs/algos.pyx
@@ -389,11 +389,11 @@ def nancorr_spearman(ndarray[float64_t, ndim=2] mat, Py_ssize_t minp=1) -> ndarr
         int64_t nobs = 0
         bint no_nans
         float64_t vx, vy, sumx, sumxx, sumyy, mean, divisor
-        const int64_t[:] labels_n, labels_nobs
+        const intp_t[:] labels_n, labels_nobs
 
     N, K = (<object>mat).shape
     # For compatibility when calling rank_1d
-    labels_n = np.zeros(N, dtype=np.int64)
+    labels_n = np.zeros(N, dtype=np.intp)
 
     # Handle the edge case where we know all results will be nan
     # to keep conditional logic inside loop simpler
@@ -451,7 +451,7 @@ def nancorr_spearman(ndarray[float64_t, ndim=2] mat, Py_ssize_t minp=1) -> ndarr
                             with gil:
                                 # We need to slice back to nobs because rank_1d will
                                 # require arrays of nobs length
-                                labels_nobs = np.zeros(nobs, dtype=np.int64)
+                                labels_nobs = np.zeros(nobs, dtype=np.intp)
                                 rankedx = rank_1d(np.array(maskedx)[:nobs],
                                                   labels=labels_nobs)
                                 rankedy = rank_1d(np.array(maskedy)[:nobs],
diff --git a/pandas/tests/frame/methods/test_cov_corr.py b/pandas/tests/frame/methods/test_cov_corr.py
index c259902ec2498..3dbf49df72558 100644
--- a/pandas/tests/frame/methods/test_cov_corr.py
+++ b/pandas/tests/frame/methods/test_cov_corr.py
@@ -100,7 +100,6 @@ def test_corr_scipy_method(self, float_frame, method):
 
     # ---------------------------------------------------------------------
 
-    @td.skip_if_no_scipy
     def test_corr_non_numeric(self, float_string_frame):
         # exclude non-numeric types
         result = float_string_frame.corr()
@@ -125,11 +124,9 @@ def test_corr_nooverlap(self, meth):
         assert rs.loc["B", "B"] == 1
         assert isna(rs.loc["C", "C"])
 
-    @td.skip_if_no_scipy
     @pytest.mark.parametrize("meth", ["pearson", "spearman"])
     def test_corr_constant(self, meth):
         # constant --> all NA
-
         df = DataFrame(
             {
                 "A": [1, 1, 1, np.nan, np.nan, np.nan],