The beIN Sports presenter, who became a household name during his lengthy spell working as Sky Sports’ leading anchor, made the surprising claim in his latest blog.
Keys believes it’s only a matter of time before Mourinho loses his job - the Portuguese coach has endured a disastrous start to the season - and feels that Roman Abramovich should look no further than Neville if the Chelsea owners opts to appoint a British coach.
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“Now, let me stress, right now there isn't a vacancy, but if there was why couldn't Chelsea go British?” Keys writes. “I’m serious. Here's a thought - what about Gary Neville? Why not?
“Gary continues to impress everybody with his thoughtful approach to the game. He's been a stunning success at Sky TV.
“He's done a terrific job. He's coached at the highest level with England. He's handled big players during his time with England. He sees a game that few others do and more often than not he's got the answers to coaching problems at most clubs. He's never shirked a tackle or a battle. He's ready to put into practice what he believes in surely?
“We need English clubs winning in the [Champions League] again, so why not give Neville a go? If winning everything as a player is a factor, and surely it counts for something, then - I ask again, why not? I honestly believe it's time Gary put down his mic and put his boots back on. I can't think of a better British candidate to manage a big club right now.
“Come on Gary, strike a blow for British coaching. Prove that our guys know what they're talking about. Give it a go.”
Neville is the best pundit on TV
Neville is widely regarded as the most astute pundit on television and, amongst a host of other projects, is also part of the England coaching staff under Roy Hodgson.
But despite his vast experience - combined with his incredible knowledge of the game - Neville is still yet to take a managerial position at a football club.
Nevile wouldn't take Chelsea job
Many believe that Neville will be a roaring success when he finally becomes a manager, but it’s doubtful that he would accept the job at Chelsea.
For starters, Neville already has commitments with both Sky and England. Secondly, the retired right-back is acutely aware of the huge risk he would be taking. Abramovich is not the sort of club owner who affords his managers time. And finally, he’s a die-hard Manchester United supporter.
Neville: 66/1 to become Chelsea manager
If, however, you believe Keys is on to something you can currently get 66/1 on Neville to become the new manager of Chelsea - the same price as Harry Redknapp, incidentally.
The current favourite is Pep Guardiola, followed by Brendan Rodgers, Carlo Ancelotti, Diego Simeone and John Terry.
Chelsea fans: would you be happy if Gary Neville became your next manager? Have your say by leaving a comment in the box below!