diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index f1f24ab7a101b..53c56fe3f6909 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -219,6 +219,7 @@ Other enhancements
 - Where possible :meth:`RangeIndex.difference` and :meth:`RangeIndex.symmetric_difference` will return :class:`RangeIndex` instead of :class:`Int64Index` (:issue:`36564`)
 - Added :meth:`Rolling.sem()` and :meth:`Expanding.sem()` to compute the standard error of mean (:issue:`26476`).
 - :meth:`Rolling.var()` and :meth:`Rolling.std()` use Kahan summation and Welfords Method to avoid numerical issues (:issue:`37051`)
+- :meth:`DataFrame.corr` and :meth:`DataFrame.cov` use Welfords Method to avoid numerical issues (:issue:`37448`)
 - :meth:`DataFrame.plot` now recognizes ``xlabel`` and ``ylabel`` arguments for plots of type ``scatter`` and ``hexbin`` (:issue:`37001`)
 - :class:`DataFrame` now supports ``divmod`` operation (:issue:`37165`)
 - :meth:`DataFrame.to_parquet` now returns a ``bytes`` object when no ``path`` argument is passed (:issue:`37105`)
diff --git a/pandas/_libs/algos.pyx b/pandas/_libs/algos.pyx
index da5ae97bb067b..620889ad39889 100644
--- a/pandas/_libs/algos.pyx
+++ b/pandas/_libs/algos.pyx
@@ -268,7 +268,8 @@ def nancorr(const float64_t[:, :] mat, bint cov=False, minp=None):
         ndarray[float64_t, ndim=2] result
         ndarray[uint8_t, ndim=2] mask
         int64_t nobs = 0
-        float64_t vx, vy, sumx, sumy, sumxx, sumyy, meanx, meany, divisor
+        float64_t vx, vy, meanx, meany, divisor, prev_meany, prev_meanx, ssqdmx
+        float64_t ssqdmy, covxy
 
     N, K = (<object>mat).shape
 
@@ -283,37 +284,29 @@ def nancorr(const float64_t[:, :] mat, bint cov=False, minp=None):
     with nogil:
         for xi in range(K):
             for yi in range(xi + 1):
-                nobs = sumxx = sumyy = sumx = sumy = 0
+                # Welford's method for the variance-calculation
+                # https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
+                nobs = ssqdmx = ssqdmy = covxy = meanx = meany = 0
                 for i in range(N):
                     if mask[i, xi] and mask[i, yi]:
                         vx = mat[i, xi]
                         vy = mat[i, yi]
                         nobs += 1
-                        sumx += vx
-                        sumy += vy
+                        prev_meanx = meanx
+                        prev_meany = meany
+                        meanx = meanx + 1 / nobs * (vx - meanx)
+                        meany = meany + 1 / nobs * (vy - meany)
+                        ssqdmx = ssqdmx + (vx - meanx) * (vx - prev_meanx)
+                        ssqdmy = ssqdmy + (vy - meany) * (vy - prev_meany)
+                        covxy = covxy + (vx - meanx) * (vy - prev_meany)
 
                 if nobs < minpv:
                     result[xi, yi] = result[yi, xi] = NaN
                 else:
-                    meanx = sumx / nobs
-                    meany = sumy / nobs
-
-                    # now the cov numerator
-                    sumx = 0
-
-                    for i in range(N):
-                        if mask[i, xi] and mask[i, yi]:
-                            vx = mat[i, xi] - meanx
-                            vy = mat[i, yi] - meany
-
-                            sumx += vx * vy
-                            sumxx += vx * vx
-                            sumyy += vy * vy
-
-                    divisor = (nobs - 1.0) if cov else sqrt(sumxx * sumyy)
+                    divisor = (nobs - 1.0) if cov else sqrt(ssqdmx * ssqdmy)
 
                     if divisor != 0:
-                        result[xi, yi] = result[yi, xi] = sumx / divisor
+                        result[xi, yi] = result[yi, xi] = covxy / divisor
                     else:
                         result[xi, yi] = result[yi, xi] = NaN
 
diff --git a/pandas/tests/frame/methods/test_cov_corr.py b/pandas/tests/frame/methods/test_cov_corr.py
index 7eeeb245534f5..6cea5abcac6d0 100644
--- a/pandas/tests/frame/methods/test_cov_corr.py
+++ b/pandas/tests/frame/methods/test_cov_corr.py
@@ -208,6 +208,25 @@ def test_corr_item_cache(self):
         assert df["A"] is ser
         assert df.values[0, 0] == 99
 
+    @pytest.mark.parametrize("length", [2, 20, 200, 2000])
+    def test_corr_for_constant_columns(self, length):
+        # GH: 37448
+        df = DataFrame(length * [[0.4, 0.1]], columns=["A", "B"])
+        result = df.corr()
+        expected = DataFrame(
+            {"A": [np.nan, np.nan], "B": [np.nan, np.nan]}, index=["A", "B"]
+        )
+        tm.assert_frame_equal(result, expected)
+
+    def test_calc_corr_small_numbers(self):
+        # GH: 37452
+        df = DataFrame(
+            {"A": [1.0e-20, 2.0e-20, 3.0e-20], "B": [1.0e-20, 2.0e-20, 3.0e-20]}
+        )
+        result = df.corr()
+        expected = DataFrame({"A": [1.0, 1.0], "B": [1.0, 1.0]}, index=["A", "B"])
+        tm.assert_frame_equal(result, expected)
+
 
 class TestDataFrameCorrWith:
     def test_corrwith(self, datetime_frame):