diff --git a/doc/whats_new/v1.4.rst b/doc/whats_new/v1.4.rst
index 73b87d260acba..f048e7f955995 100644
--- a/doc/whats_new/v1.4.rst
+++ b/doc/whats_new/v1.4.rst
@@ -24,6 +24,13 @@ random sampling procedures.
   and has been fixed.
   :pr:`26416` by :user:`Yang Tao <mchikyt3>`.
 
+- |Fix| Ridge models with `solver='sparse_cg'` may have slightly different
+  results with scipy>=1.12, because of an underlying change in the scipy solver
+  (see `scipy#18488 <https://github.com/scipy/scipy/pull/18488>`_ for more
+  details)
+  :pr:`26814` by :user:`Loïc Estève <lesteve>`
+
+
 Changes impacting all modules
 -----------------------------
 
diff --git a/sklearn/linear_model/_ridge.py b/sklearn/linear_model/_ridge.py
index 63c02185fe4a1..0258a379b8852 100644
--- a/sklearn/linear_model/_ridge.py
+++ b/sklearn/linear_model/_ridge.py
@@ -33,6 +33,7 @@
 )
 from ..utils._param_validation import Interval, StrOptions, validate_params
 from ..utils.extmath import row_norms, safe_sparse_dot
+from ..utils.fixes import _sparse_linalg_cg
 from ..utils.sparsefuncs import mean_variance_axis
 from ..utils.validation import _check_sample_weight, check_is_fitted
 from ._base import LinearClassifierMixin, LinearModel, _preprocess_data, _rescale_data
@@ -105,7 +106,7 @@ def _mv(x):
             C = sp_linalg.LinearOperator(
                 (n_samples, n_samples), matvec=mv, dtype=X.dtype
             )
-            coef, info = sp_linalg.cg(C, y_column, tol=tol, atol="legacy")
+            coef, info = _sparse_linalg_cg(C, y_column, rtol=tol)
             coefs[i] = X1.rmatvec(coef)
         else:
             # linear ridge
@@ -114,9 +115,7 @@ def _mv(x):
             C = sp_linalg.LinearOperator(
                 (n_features, n_features), matvec=mv, dtype=X.dtype
             )
-            coefs[i], info = sp_linalg.cg(
-                C, y_column, maxiter=max_iter, tol=tol, atol="legacy"
-            )
+            coefs[i], info = _sparse_linalg_cg(C, y_column, maxiter=max_iter, rtol=tol)
 
         if info < 0:
             raise ValueError("Failed with error code %d" % info)
diff --git a/sklearn/utils/fixes.py b/sklearn/utils/fixes.py
index d33b638358157..ba5ce3c35d07f 100644
--- a/sklearn/utils/fixes.py
+++ b/sklearn/utils/fixes.py
@@ -15,6 +15,7 @@
 
 import numpy as np
 import scipy
+import scipy.sparse.linalg
 import scipy.stats
 import threadpoolctl
 
@@ -109,6 +110,19 @@ def _mode(a, axis=0):
     return scipy.stats.mode(a, axis=axis)
 
 
+# TODO: Remove when Scipy 1.12 is the minimum supported version
+if sp_base_version >= parse_version("1.12.0"):
+    _sparse_linalg_cg = scipy.sparse.linalg.cg
+else:
+
+    def _sparse_linalg_cg(A, b, **kwargs):
+        if "rtol" in kwargs:
+            kwargs["tol"] = kwargs.pop("rtol")
+        if "atol" not in kwargs:
+            kwargs["atol"] = "legacy"
+        return scipy.sparse.linalg.cg(A, b, **kwargs)
+
+
 ###############################################################################
 # Backport of Python 3.9's importlib.resources
 # TODO: Remove when Python 3.9 is the minimum supported version