"I don't think you can give it to someone else, otherwise there would be mutiny," former United defender Neville, who was Solskjaer's team mate from 1996 to 2007, told Sky Sports.
"The fans were singing for him. I think it's very close now -- I can't see the club going away from it."
Injuries forced the Norwegian to make all three substitutions before half-time in Sunday's 0-0 league draw with Liverpool but Old Trafford was in full voice throughout the 90 minutes, singing the name of their manager.
"Three months ago they were singing 'attack, attack, attack' and today they were singing Ole Gunnar Solskjaer's name for all of the second half," Neville added.
"The mood has completely transformed - the atmosphere was one of the best I've seen at Old Trafford for years and it was 0-0! If Manchester United had scored it would have been incredible."
United, who trail fourth-placed Arsenal by a point in the league table, next visit Crystal Palace on Wednesday.
(Reporting by Hardik Vyas in Bengaluru; editing by Sudipto Ganguly)
