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Mourinho, 58, succeeded Louis van Gaal at Old Trafford in 2016 and led United to a League Cup and Europa League during his first season in charge.
The Special One’s second season at United ended without silverware, but Mourinho led the Red Devils to second place in the Premier League table.
United’s second-placed finish was their highest league position since Sir Alex Ferguson announced his retirement in 2013.
Mourinho’s two-and-a-half-year spell in charge of United came to an end in December 2018, with Ole Gunnar Solskjaer succeeding the Special One at Old Trafford.
Speaking after Liverpool’s thumping 5-0 win over United on Sunday, Neville claimed he was “disappointed” during the eras of Mourinho and Van Gaal at the Red Devils.
“I've seen Jose [Mourinho] and [Louis] van Gaal come in and I was disappointed,” he said on Sky Sports.
“The reason the club will go at the end of the season, they won't bring a hitman in. They will stick with Ole and he has to win something this season. Any other club, he'd be in trouble now.
“Jose was the best manager in the world and couldn't do it. I wouldn't bring [Antonio] Conte into United. He's not a fit for United.
“I might be wrong, but I don't think that he is. I think it would be wrong to sack Ole now. I sacked a manager after six games last season. It was the wrong thing to do, I should've let things breathe.”
Keys has now hit out at the Sky Sports pundit over his comments on Mourinho, who led United to second with a “bang ordinary squad.”
The former Sky Sports presenter wrote in a blog post: “Fergie, of course, chose David Moyes -- and chose to go missing when Moyes needed his backing.
“LVG and Mourinho were Mr Ed’s, who wanted to prove he could make big decisions without Fergie.
“By the way, I’m sorry Gary -- but to say Mourinho was a failure at United is ridiculous.
“He finished 2nd with a bang ordinary squad and delivered two trophies. I’d say being run out of Spain after 6 months was failure.”
Mourinho, who went on to manage Tottenham and current side Roma after United, says one of his top achievements was leading the Red Devils to second in the Premier League.
Speaking on beIN SPORTS in 2019, he said: “If I tell you, for example, that I consider one of the best jobs of my career was to finish second with Man United in the Premier League, you will say, ‘This guy is crazy. He won 25 titles and he is saying that a second position was one of his best achievements?’
“I keep saying this because people don't know what is going on behind the scenes.
“And sometimes we, in this side of the cameras, analyse things with a different perspective.”