Reports at the weekend suggested that the club planned to sack Mourinho regardless of their result against Newcastle United, although United later rejected that speculation.
The Red Devils conceded twice in the opening 10 minutes against the Magpies, but launched a stirring second-half comeback to win 3-2 and Neville is confident that Mourinho still has the ability to turn things around.
"I think Jose is still someone that I look up to and admire because managers go through tough times. I've been sacked twice and it hurts when people level things at you, in defeat or at bad times, it can get personal and maybe some of it has got personal with Jose," Neville told Sky Sports News.
"I agree with Jose I think there is a bit of a manhunt out there - everyone seems to be looking at him. I think they've got the right man who can handle that sort of situation. My feeling when I think of the great Manchester United teams I think of the togetherness of the whole club. The directors coming down in the dressing room after we had won European Cup finals, FA Cup finals - we shared the success. We used to celebrate with the supporters on the bus tours of the city.
"It's all or nothing. You can't go fighting a war on your own, you've got to fight it together and I think that would be the biggest thing about Saturday - all of a sudden everyone came together. I hate what's happening with the criticism that the club are getting because it hurts a lot, but Saturday was a day that I think we could all enjoy as supporters. If you had been a Manchester United fan you would say that was quite normal - 2-0 down and coming back to win 3-2."
Saturday's victory over Newcastle was their first at Old Trafford in any competition for almost two months.