The Portuguese is heavily expected to replace Louis van Gaal at Manchester United.
They did, however, win the FA Cup on Saturday by beating Crystal Palace 2-1 after extra time, but according to Sky Sports it will not be enough to save his job.
With Euro 2016 starting in 19 days, it may be easier for the former Chelsea boss to make a start on his summer business before the tournament kicks off, and Neville believes that Mourinho will "demand" that his targets join the club.
"Don't be surprised if United in the next month, and particularly before the Euros, sign three or four players and when he gets to pre-season he has his squad already in place," he's quoted as saying by The Express.
"He knows who he wants and at the start he will demand United sign those players for United to become championship challengers again."
Hiring Mourinho surely represents a good move by United, if it materialises. The 53-year-old was sacked by Chelsea in December after his team - then the reigning Premier League champions - appeared to stop playing for him.
But with two Champions League titles and three Premier League crowns on his CV, an available Mourinho simply cannot be sniffed at.