The former Australia international believes the current boss must guide his side to victories over Olympiakos and Manchester City or face the sack
United's 3-0 defeat to rivals Liverpool on Sunday increased the pressure on Moyes, with United now 12 points adrift of the Premier League top four and 18 behind leaders Chelsea.
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United then travel to West Ham on Saturday before hosting fierce rivals Manchester City on Tuesday, and Bosnich has suggested the outcome of the three games will determine Moyes' future, despite the Scot claiming on Tuesday that he has the backing of the Old Trafford board.
"David Moyes is a wonderful man but things have gone from bad to worse," he told talkSPORT. "They've got big problems. You can see it out on the pitch.
"The two games [against Olympiakos and Manchester City] are huge.
"The West Ham match will take care of itself, but the results against those two will determine whether David Moyes stays at the club.
"If we see another two abject performances, especially in the derby, I don't think the board will have any choice but to appoint someone else."