diff --git a/python-package/lightgbm/engine.py b/python-package/lightgbm/engine.py
index 646c1ec8505d..2bca6cbdb97b 100644
--- a/python-package/lightgbm/engine.py
+++ b/python-package/lightgbm/engine.py
@@ -178,7 +178,7 @@ def train(
     params["num_iterations"] = num_boost_round
     # show deprecation warning only for early stop argument, setting early stop via global params should still be possible
     if early_stopping_rounds is not None and early_stopping_rounds > 0:
-        _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'early_stopping()' callback via 'callbacks' argument instead.")
     for alias in _ConfigAliases.get("early_stopping_round"):
         if alias in params:
@@ -237,9 +237,9 @@ def train(
 
     # Most of legacy advanced options becomes callbacks
     if verbose_eval != "warn":
-        _log_warning("'verbose_eval' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'verbose_eval' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'print_evaluation()' callback via 'callbacks' argument instead.")
-    if verbose_eval == "warn":
+    else:
         if callbacks:  # assume user has already specified print_evaluation callback
             verbose_eval = False
         else:
@@ -253,12 +253,12 @@ def train(
         callbacks.add(callback.early_stopping(early_stopping_rounds, first_metric_only, verbose=bool(verbose_eval)))
 
     if learning_rates is not None:
-        _log_warning("'learning_rates' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'learning_rates' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'reset_parameter()' callback via 'callbacks' argument instead.")
         callbacks.add(callback.reset_parameter(learning_rate=learning_rates))
 
     if evals_result is not None:
-        _log_warning("'evals_result' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'evals_result' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'record_evaluation()' callback via 'callbacks' argument instead.")
         callbacks.add(callback.record_evaluation(evals_result))
 
@@ -575,7 +575,7 @@ def cv(params, train_set, num_boost_round=100,
             num_boost_round = params.pop(alias)
     params["num_iterations"] = num_boost_round
     if early_stopping_rounds is not None and early_stopping_rounds > 0:
-        _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'early_stopping()' callback via 'callbacks' argument instead.")
     for alias in _ConfigAliases.get("early_stopping_round"):
         if alias in params:
@@ -618,7 +618,7 @@ def cv(params, train_set, num_boost_round=100,
     if early_stopping_rounds is not None and early_stopping_rounds > 0:
         callbacks.add(callback.early_stopping(early_stopping_rounds, first_metric_only, verbose=False))
     if verbose_eval is not None:
-        _log_warning("'verbose_eval' argument is deprecated and will be removed in 4.0.0 release. "
+        _log_warning("'verbose_eval' argument is deprecated and will be removed in a future release of LightGBM. "
                      "Pass 'print_evaluation()' callback via 'callbacks' argument instead.")
     if verbose_eval is True:
         callbacks.add(callback.print_evaluation(show_stdv=show_stdv))
diff --git a/python-package/lightgbm/sklearn.py b/python-package/lightgbm/sklearn.py
index 332f971b4c3d..ba1c447a16f7 100644
--- a/python-package/lightgbm/sklearn.py
+++ b/python-package/lightgbm/sklearn.py
@@ -711,7 +711,7 @@ def _get_meta_data(collection, name, i):
             init_model = init_model.booster_
 
         if early_stopping_rounds is not None and early_stopping_rounds > 0:
-            _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in 4.0.0 release. "
+            _log_warning("'early_stopping_rounds' argument is deprecated and will be removed in a future release of LightGBM. "
                          "Pass 'early_stopping()' callback via 'callbacks' argument instead.")
             params['early_stopping_rounds'] = early_stopping_rounds
 
@@ -721,9 +721,9 @@ def _get_meta_data(collection, name, i):
             callbacks = copy.deepcopy(callbacks)
 
         if verbose != 'warn':
-            _log_warning("'verbose' argument is deprecated and will be removed in 4.0.0 release. "
+            _log_warning("'verbose' argument is deprecated and will be removed in a future release of LightGBM. "
                          "Pass 'print_evaluation()' callback via 'callbacks' argument instead.")
-        if verbose == 'warn':
+        else:
             if callbacks:  # assume user has already specified print_evaluation callback
                 verbose = False
             else:
@@ -748,8 +748,14 @@ def _get_meta_data(collection, name, i):
 
         if evals_result:
             self._evals_result = evals_result
+        else:  # reset after previous call to fit()
+            self._evals_result = None
+
+        if self._Booster.best_iteration != 0:
+            self._best_iteration = self._Booster.best_iteration
+        else:  # reset after previous call to fit()
+            self._best_iteration = None
 
-        self._best_iteration = self._Booster.best_iteration
         self._best_score = self._Booster.best_score
 
         self.fitted_ = True
@@ -810,14 +816,14 @@ def n_features_in_(self):
 
     @property
     def best_score_(self):
-        """:obj:`dict`: The best score of fitted model."""
+        "":obj:`dict`: The best score of fitted model."""
         if self._n_features is None:
             raise LGBMNotFittedError('No best_score found. Need to call fit beforehand.')
         return self._best_score
 
     @property
     def best_iteration_(self):
-        """:obj:`int`: The best iteration of fitted model if ``early_stopping()`` callback has been specified."""
+        """:obj:`int` or :obj:`None`: The best iteration of fitted model if ``early_stopping()`` callback has been specified."""
         if self._n_features is None:
             raise LGBMNotFittedError('No best_iteration found. Need to call fit with early_stopping callback beforehand.')
         return self._best_iteration
diff --git a/tests/python_package_test/test_dask.py b/tests/python_package_test/test_dask.py
index 11e512ffd996..2f84933cfd37 100644
--- a/tests/python_package_test/test_dask.py
+++ b/tests/python_package_test/test_dask.py
@@ -916,7 +916,7 @@ def test_eval_set_no_early_stopping(task, output, eval_sizes, eval_names_prefix,
 
             # check that early stopping was not applied.
             assert dask_model.booster_.num_trees() == model_trees
-            assert dask_model.best_iteration_ == 0
+            assert dask_model.best_iteration_ is None
 
             # checks that evals_result_ and best_score_ contain expected data and eval_set names.
             evals_result = dask_model.evals_result_