diff --git a/smt/surrogate_models/krg_based.py b/smt/surrogate_models/krg_based.py
index d8c853bb8..c17bcd856 100644
--- a/smt/surrogate_models/krg_based.py
+++ b/smt/surrogate_models/krg_based.py
@@ -307,6 +307,14 @@ def set_training_values(
             Matrix specifying which of the design variables is acting in a hierarchical design space
         """
         super().set_training_values(xt, yt, name=name)
+        if self.ny > 1:
+            warnings.warn(
+                "Kriging-based surrogate is not intended to handle multiple "
+                f"training output data (yt dim should be 1, got {self.ny}). "
+                "The quality of the resulting surrogate might not be as good as "
+                "if each training output is used separately to build a dedicated surrogate. "
+                "This warning might become a hard error in future SMT versions."
+            )
         if is_acting is not None:
             self.is_acting_points[name] = is_acting
 
@@ -397,6 +405,7 @@ def _new_train(self):
         # Sampling points X and y
         X = self.training_points[None][0][0]
         y = self.training_points[None][0][1]
+
         # Get is_acting status from design space model if needed (might correct training points)
         is_acting = self.is_acting_points.get(None)
         if is_acting is None and not self.is_continuous:
diff --git a/smt/surrogate_models/tests/test_krg_based.py b/smt/surrogate_models/tests/test_krg_based.py
index 47fe9accd..a1c2652ae 100644
--- a/smt/surrogate_models/tests/test_krg_based.py
+++ b/smt/surrogate_models/tests/test_krg_based.py
@@ -40,6 +40,11 @@ def test_theta0_erroneous_init(self):
         krg.set_training_values(np.array([[1, 2, 3]]), np.array([[1]]))  # erroneous
         self.assertRaises(ValueError, krg._check_param)
 
+    def test_multiple_training_outputs_warning(self):
+        krg = KrgBased()
+        with self.assertWarns(UserWarning):
+            krg.set_training_values(np.array([[1, 2, 3]]), np.array([[1, 1]]))
+
     def test_less_almost_squar_exp(self):
         nobs = 50  # number of obsertvations
         np.random.seed(0)  # a seed for reproducibility