The origin to the idea that Montgomery claimed to have rescued an incompetent US command at the time of the Ardennes offensive, is almost entirely due to a cleverly produced piece of German propaganda that was put out in the guise of a BBC announcement.
You can read all about it in this essay (2020) by staff of the National Archives in Britain.
Initially all concerned - including American forces all over Europe - fell for what appeared to be - a BBC report. Unfortunately the matter was taken up by some of the more jingoistic press in Britain - notably the Daily Mail - who also fell for the same "fake news".
From what I have read elsewhere, there may be some evidence that Montgomery warned Eisenhower, a couple of weeks before the German offensive, that it was likely to happen due to the large gap in the Ardennes front, held only by troops that were "resting".
Remember the British commanders had had greater experience of battle - from WW1 - than the Americans had had. But I strongly suspect that Monty, especially in Italy, had not withheld his contempt of some US commanders he saw as "green-behind-the-ears" that he was not taken as seriously as he should have been on this occasion. This was surely an important lesson for both sides.