diff --git a/doc/whats_new/v1.0.rst b/doc/whats_new/v1.0.rst
index 84a86dc688aa9..25d6e95064a09 100644
--- a/doc/whats_new/v1.0.rst
+++ b/doc/whats_new/v1.0.rst
@@ -43,6 +43,12 @@ Fixed models
   between sparse and dense input. :pr:`21195`
   by :user:`Jérémie du Boisberranger <jeremiedbb>`.
 
+:mod:`sklearn.linear_model`
+...........................
+
+- |Fix| Improves stability of :class:`linear_model.LassoLars` for different
+  versions of openblas. :pr:`21340` by `Thomas Fan`_.
+
 :mod:`sklearn.neighbors`
 ........................
 
@@ -677,7 +683,7 @@ Changelog
   Dupre la Tour`_.
 
 - |Fix| Decrease the numerical default tolerance in the lobpcg call
-  in :func:`manifold.spectral_embedding` to prevent numerical instability. 
+  in :func:`manifold.spectral_embedding` to prevent numerical instability.
   :pr:`21194` by :user:`Andrew Knyazev <lobpcg>`.
 
 :mod:`sklearn.metrics`
diff --git a/sklearn/linear_model/_least_angle.py b/sklearn/linear_model/_least_angle.py
index c9137cbf056ec..351aa20f549c2 100644
--- a/sklearn/linear_model/_least_angle.py
+++ b/sklearn/linear_model/_least_angle.py
@@ -544,6 +544,7 @@ def _lars_path_solver(
             sys.stdout.flush()
 
     tiny32 = np.finfo(np.float32).tiny  # to avoid division by 0 warning
+    cov_precision = np.finfo(Cov.dtype).precision
     equality_tolerance = np.finfo(np.float32).eps
 
     if Gram is not None:
@@ -726,6 +727,10 @@ def _lars_path_solver(
             # orthogonal (QR) decomposition of X
             corr_eq_dir = np.dot(Gram[:n_active, n_active:].T, least_squares)
 
+        # Explicit rounding can be necessary to avoid `np.argmax(Cov)` yielding
+        # unstable results because of rounding errors.
+        np.around(corr_eq_dir, decimals=cov_precision, out=corr_eq_dir)
+
         g1 = arrayfuncs.min_pos((C - Cov) / (AA - corr_eq_dir + tiny32))
         if positive:
             gamma_ = min(g1, C / AA)