diff --git a/sklearn/metrics/_ranking.py b/sklearn/metrics/_ranking.py
index 0cdead9233898..28c89ef6357e4 100644
--- a/sklearn/metrics/_ranking.py
+++ b/sklearn/metrics/_ranking.py
@@ -1019,7 +1019,8 @@ def roc_curve(
     thresholds : ndarray of shape = (n_thresholds,)
         Decreasing thresholds on the decision function used to compute
         fpr and tpr. `thresholds[0]` represents no instances being predicted
-        and is arbitrarily set to `max(y_score) + 1`.
+        and is set to the next representable value above ``max(y_score)`` for
+        floating scores (or ``max(y_score) + 1`` otherwise).
 
     See Also
     --------
@@ -1083,7 +1084,11 @@ def roc_curve(
     # to make sure that the curve starts at (0, 0)
     tps = np.r_[0, tps]
     fps = np.r_[0, fps]
-    thresholds = np.r_[thresholds[0] + 1, thresholds]
+    if np.issubdtype(thresholds.dtype, np.floating):
+        prepend_thresh = np.nextafter(thresholds[0], np.inf, dtype=thresholds.dtype)
+    else:
+        prepend_thresh = thresholds[0] + 1
+    thresholds = np.r_[prepend_thresh, thresholds]
 
     if fps[-1] <= 0:
         warnings.warn(
diff --git a/sklearn/metrics/tests/test_ranking.py b/sklearn/metrics/tests/test_ranking.py
index f38f118c38c0a..89cb8b12ae3d5 100644
--- a/sklearn/metrics/tests/test_ranking.py
+++ b/sklearn/metrics/tests/test_ranking.py
@@ -216,6 +216,7 @@ def test_roc_curve_end_points():
     assert fpr.shape == thr.shape
 
 
+
 def test_roc_returns_consistency():
     # Test whether the returned threshold matches up with tpr
     # make small toy dataset
@@ -2199,3 +2200,46 @@ def test_ranking_metric_pos_label_types(metric, classes):
         assert not np.isnan(metric_1).any()
         assert not np.isnan(metric_2).any()
         assert not np.isnan(thresholds).any()
+
+
+def test_roc_curve_thresholds_probabilities_below_one():
+    # Probabilistic scores below 1.0 should prepend the smallest greater float
+    y_true = np.array([0, 0, 1, 1])
+    y_score = np.array([0.75, 0.2, 0.33, 0.9], dtype=np.float64)
+
+    fpr, tpr, thresholds = roc_curve(y_true, y_score, drop_intermediate=False)
+
+    max_score = y_score.max()
+    expected_prepend = np.nextafter(max_score, np.inf)
+
+    assert thresholds[0] == pytest.approx(expected_prepend)
+    assert thresholds[0] > max_score
+    assert np.all(np.diff(thresholds) <= 0)
+    assert thresholds.shape == fpr.shape == tpr.shape
+
+
+def test_roc_curve_thresholds_probability_one():
+    # The prepend threshold for a max score of 1.0 should be nextafter(1.0)
+    y_true = np.array([0, 1, 0, 1])
+    y_score = np.array([1.0, 0.6, 0.4, 0.8], dtype=np.float64)
+
+    fpr, tpr, thresholds = roc_curve(y_true, y_score, drop_intermediate=False)
+
+    expected_prepend = np.nextafter(1.0, np.inf, dtype=thresholds.dtype)
+
+    assert thresholds[0] == pytest.approx(expected_prepend)
+    assert thresholds[1] == pytest.approx(1.0)
+    assert np.all(np.diff(thresholds) <= 0)
+    assert thresholds.shape == fpr.shape == tpr.shape
+
+
+def test_roc_curve_thresholds_integer_scores():
+    # Non-probability integer scores should continue to prepend max(score) + 1
+    y_true = np.array([0, 1, 0, 1])
+    y_score = np.array([2, 3, 1, 0], dtype=np.int64)
+
+    fpr, tpr, thresholds = roc_curve(y_true, y_score, drop_intermediate=False)
+
+    assert thresholds[0] == y_score.max() + 1
+    assert np.all(np.diff(thresholds) <= 0)
+    assert thresholds.shape == fpr.shape == tpr.shape