diff --git a/sklearn/multioutput.py b/sklearn/multioutput.py
index fbacdc79aa140..fe846492bddda 100644
--- a/sklearn/multioutput.py
+++ b/sklearn/multioutput.py
@@ -660,6 +660,8 @@ def fit(self, X, Y, **fit_params):
 
         del Y_pred_chain
 
+        routed_params = process_routing(obj=self, method="fit", other_params=fit_params)
+
         for chain_idx, estimator in enumerate(self.estimators_):
             message = self._log_message(
                 estimator_idx=chain_idx + 1,
@@ -668,7 +670,12 @@ def fit(self, X, Y, **fit_params):
             )
             y = Y[:, self.order_[chain_idx]]
             with _print_elapsed_time("Chain", message):
-                estimator.fit(X_aug[:, : (X.shape[1] + chain_idx)], y, **fit_params)
+                estimator.fit(
+                    X_aug[:, : (X.shape[1] + chain_idx)],
+                    y,
+                    **routed_params.estimator.fit,
+                )
+
             if self.cv is not None and chain_idx < len(self.estimators_) - 1:
                 col_idx = X.shape[1] + chain_idx
                 cv_result = cross_val_predict(
@@ -831,7 +838,7 @@ class labels for each estimator in the chain.
            [0.0321..., 0.9935..., 0.0625...]])
     """
 
-    def fit(self, X, Y):
+    def fit(self, X, Y, **fit_params):
         """Fit the model to data matrix X and targets Y.
 
         Parameters
@@ -842,6 +849,11 @@ def fit(self, X, Y):
         Y : array-like of shape (n_samples, n_classes)
             The target values.
 
+        **fit_params : dict of string -> object
+            Parameters passed to the `fit` method of each step.
+
+            .. versionadded:: 1.2
+
         Returns
         -------
         self : object
@@ -849,7 +861,7 @@ def fit(self, X, Y):
         """
         self._validate_params()
 
-        super().fit(X, Y)
+        super().fit(X, Y, **fit_params)
         self.classes_ = [
             estimator.classes_ for chain_idx, estimator in enumerate(self.estimators_)
         ]
@@ -919,6 +931,24 @@ def decision_function(self, X):
 
         return Y_decision
 
+    def get_metadata_routing(self):
+        """Get metadata routing of this object.
+
+        Please check :ref:`User Guide <metadata_routing>` on how the routing
+        mechanism works.
+
+        Returns
+        -------
+        routing : MetadataRouter
+            A :class:`~utils.metadata_routing.MetadataRouter` encapsulating
+            routing information.
+        """
+        router = MetadataRouter(owner=self.__class__.__name__).add(
+            estimator=self.base_estimator,
+            method_mapping=MethodMapping().add(callee="fit", caller="fit"),
+        )
+        return router
+
     def _more_tags(self):
         return {"_skip_test": True, "multioutput_only": True}
 
@@ -1046,5 +1076,27 @@ def fit(self, X, Y, **fit_params):
         super().fit(X, Y, **fit_params)
         return self
 
+    def get_metadata_routing(self):
+        """Get metadata routing of this object.
+
+        Please check :ref:`User Guide <metadata_routing>` on how the routing
+        mechanism works.
+
+        Returns
+        -------
+        routing : MetadataRouter
+            A :class:`~utils.metadata_routing.MetadataRouter` encapsulating
+            routing information.
+        """
+        router = (
+            MetadataRouter(owner=self.__class__.__name__)
+            .add(
+                estimator=self.base_estimator,
+                method_mapping=MethodMapping().add(callee="fit", caller="fit"),
+            )
+            .warn_on(child="estimator", method="fit", params=None)
+        )
+        return router
+
     def _more_tags(self):
         return {"multioutput_only": True}
diff --git a/sklearn/tests/test_metaestimators_metadata_routing.py b/sklearn/tests/test_metaestimators_metadata_routing.py
index 603e3ae1e8d99..9cc21135cf225 100644
--- a/sklearn/tests/test_metaestimators_metadata_routing.py
+++ b/sklearn/tests/test_metaestimators_metadata_routing.py
@@ -7,7 +7,12 @@
 from sklearn.base import RegressorMixin, ClassifierMixin, BaseEstimator
 from sklearn.calibration import CalibratedClassifierCV
 from sklearn.exceptions import UnsetMetadataPassedError
-from sklearn.multioutput import MultiOutputRegressor, MultiOutputClassifier
+from sklearn.multioutput import (
+    MultiOutputRegressor,
+    MultiOutputClassifier,
+    ClassifierChain,
+    RegressorChain,
+)
 from sklearn.utils.metadata_routing import MetadataRouter
 from sklearn.tests.test_metadata_routing import (
     record_metadata,
@@ -181,6 +186,24 @@ def predict_log_proba(self, X, sample_weight="default", metadata="default"):
         "warns_on": {"fit": ["sample_weight", "metadata"]},
         "preserves_metadata": False,
     },
+    {
+        "metaestimator": ClassifierChain,
+        "estimator_name": "base_estimator",
+        "estimator": ConsumingClassifier,
+        "X": X,
+        "y": y_multi,
+        "routing_methods": ["fit"],
+        "warns_on": {},
+    },
+    {
+        "metaestimator": RegressorChain,
+        "estimator_name": "base_estimator",
+        "estimator": ConsumingRegressor,
+        "X": X,
+        "y": y_multi,
+        "routing_methods": ["fit"],
+        "warns_on": {"fit": ["sample_weight", "metadata"]},
+    },
 ]
 """List containing all metaestimators to be tested and their settings
 
diff --git a/sklearn/tests/test_multioutput.py b/sklearn/tests/test_multioutput.py
index f89734833d019..416d82a1b47e9 100644
--- a/sklearn/tests/test_multioutput.py
+++ b/sklearn/tests/test_multioutput.py
@@ -28,6 +28,7 @@
     Ridge,
     SGDClassifier,
     SGDRegressor,
+    QuantileRegressor,
 )
 from sklearn.metrics import jaccard_score, mean_squared_error
 from sklearn.model_selection import GridSearchCV, train_test_split
@@ -646,7 +647,7 @@ def fit(self, X, y, **fit_params):
             self.sample_weight_ = fit_params["sample_weight"]
             super().fit(X, y, **fit_params)
 
-    model = RegressorChain(MySGD())
+    model = RegressorChain(MySGD().set_fit_request(sample_weight=True))
 
     # Fitting with params
     fit_param = {"sample_weight": weight}
@@ -655,6 +656,16 @@ def fit(self, X, y, **fit_params):
     for est in model.estimators_:
         assert est.sample_weight_ is weight
 
+    # TODO(1.4): Remove check for FutureWarning
+    # Test that the existing behavior works and raises a FutureWarning
+    # when the underlying estimator used has a sample_weight parameter
+    # defined in it's fit method.
+    model = RegressorChain(QuantileRegressor())
+    fit_param = {"sample_weight": weight}
+
+    with pytest.warns(FutureWarning):
+        model.fit(X, y, **fit_param)
+
 
 @pytest.mark.parametrize(
     "MultiOutputEstimator, Estimator",