United's 3-0 defeat to rivals Liverpool at Old Trafford on Sunday increased the pressure on Moyes, with United now 12 points adrift of the Premier League top four and 18 behind leaders Chelsea.
The reigning English champions face Olympiacos in the UEFA Champions League on Wednesday, having been beaten 2-0 in the first leg of their last 16 tie.
Bosnich believes elimination from the competition could move Moyes closer to the exit.
United then travel to West Ham on Saturday before hosting fierce rivals Manchester City on Tuesday, and Bosnich has suggested the outcome of the derby is likely to have huge significance.
"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 Olympiacos 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."
