However, the former Red Devils and England defender believes the club should continue sounding out other options for the summer in case the Norwegian’s early momentum can’t be maintained.
After replacing sacked boss Jose Mourinho in December on a caretaker basis, Solskjaer led United to eight consecutive victories before their draw with Burnley in the Premier League on Tuesday.
"It will be hard to ignore him," Neville told Sky Sports News.
"Ultimately, what Manchester United were going to do six weeks ago was to scour Europe for one of the best managers in the world to take over in the summer.
“And they should still continue to do that, because you don't know what's going to happen in the next three or four months.
“But if Ole keeps on doing what he's doing - in fact, anything like he's doing - then there's a real dilemma in March or April in terms of what the club's plans potentially were, in terms of who would have been brought in and then obviously in terms of what they may become.
"Because if Ole has the fans on his side, the players on his side and most importantly he has got the results to the level they've got to now, then it will be incredibly difficult for the board at Manchester United to look anywhere else
"However, if they've got one of the best managers in the world secured as well, then there's that dilemma as to whether they go with that. They have tried that the last couple of times with Louis van Gaal and Jose Mourinho and it didn't work out.
"So I think that Ole is in with a real shout, particularly with the results so outstanding. The football is better and the fans feel a lot better over there at Old Trafford. The judgement should be made in March, April, May and not really for today.
“But he has a far better than he had six weeks ago."
