The Portuguese coach extended his contract at Old Trafford until 2020 earlier this year but was left frustrated in his efforts to sign a new centre-back prior to the window's closure.
Man United fell to a humbling 3-2 defeat at Brighton & Hove Albion on Sunday which prompted much criticism from fans and pundits, as well as yielding questions over Mourinho's future at the club.
"The minute that [executive vice-chairman Ed Woodward] gave Mourinho a contract extension - which some would say was unnecessary part way through a season - but believing in a manager, showing the faith, he had to then buy him the centre-backs," Neville told Sky Sports News.
"Irrespective of the manager, Manchester United need new centre-backs. My view is that [Harry] Maguire and [Toby] Alderweireld would have been good options. If they cost £120m to bring them in, then that's what you have to do.
"My view would be that he's got to stay until the very end of that contract and be backed until the end of that contract if you're going to give a contract extension as the CEO of the club.
"At the moment, there is definitely an angst there between the club and the manager. Don't get me wrong - the club has supported him. But you can't three quarters build a house - you have to finish the job. United need serious football people to guide what they are doing."
United's next match sees them host Tottenham Hotspur at Old Trafford next Monday night.