Arsene Wenger's future at Arsenal is in doubt following the Gunners' 5-1 defeat to Bayern Munich on Wednesday.
The Frenchman has insisted he will not retire this summer, even if it means managing a club other than Arsenal.
But Murphy expects Wenger and the Gunners' top scorer Sanchez to leave the club.
"When enough fans want a manager out, they usually get their way," Murphy told the London Evening Standard.
Danny Murphy expects Alexis Sanchez and Arsene Wenger to both leave Arsenal in the summer
"After what has happened to Arsenal this week, I would be amazed if Arsene Wenger was still there at the start of next season.
"A club as big as Arsenal need to be succeeding in the Premier League and the Champions League and they are not close to doing either.
"Their challenge often falls away at this time of year and it cannot be a coincidence that it keeps happening.
"It’s not only Wenger who I expect to leave, either. When you watch Alexis Sanchez, the vibe you get is that he wants more.
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"He wants to challenge for league titles and the Champions League and, when he joined Arsenal in 2014, he probably believed that would happen but they haven’t made progress.
"Sanchez’s contract expires at the end of next season and the club are desperate for him to sign a new one.
"But I wouldn’t blame him if he now thinks: “I am better than this”.
"There will be bigger, more successful clubs than Arsenal who look at Sanchez and think: “He will do for us”."