The Scotsman's future has come into question following Sunday's 3-0 defeat to arch rivals Liverpool at Old Trafford.
Bosnich believes that if the club get knocked out of the Champions League by Olympiacos on Wednesday and lose to Manchester City after their clash against West Ham United, Moyes could be shown the door.
"David Moyes is a wonderful man but things have gone from bad to worse," Bosnich told talkSPORT. "They've got big problems. You can see it out on the pitch.
"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."
United are currently seventh in the Premier League table, 12 points adrift of a Champions League place.