The 28-year-old played under the Portuguese tactician at Stamford Bridge and had a key role in helping the Blues win the title in the 2014/15 season. Mourinho was shown the exit door in December 2015 and was replaced by Antonio Conte at Chelsea.
Mourinho sanctioned Morgan Schneiderlin and Bastian Schweinsteiger's sale last season, while handing a one-year contract extension to Michael Carrick, who will turn 36 at the end of this month. Neville believes signing a holding midfielder should be a priority for United and has advised his former club to secure Matic's signature.
"United need another holding midfield player. Michael Carrick has got one more year but he can't play every single week so they do need to have that player who can come in and do the same job - and Matic would be able to do that," Neville was quoted as saying by Sky Sports.
"Morgan Schneiderlin was let go earlier in the season and Bastian Schweinsteiger didn't get a game, so you're talking about there needing to be a replacement for them and ultimately Matic would fit the bill."
Matic was vital alongside N'Golo Kante in the west London club's last title-winning campaign. However, the Serbian international's employers will be ready to sanction his sale as the former Juventus and Italy manager has set his sights on signing Tiemoue Bakayoko. Chelsea are edging closer towards completing the deal for the AS Monaco star.
Neville is surprised by the recent reports suggesting that Chelsea will allow Matic to leave the club this summer after they complete the deal for Bakayoko.
"It's a strange one for Chelsea to let him go because I thought he was really good for them with Kante. I think they need to strengthen, so I'm not sure why they're letting him and they must have a replacement," he stressed.
United managed to beat Chelsea in completing the signing of Romelu Lukaku. The former Everton striker was the Red Devils' second summer signing after the arrival of the Swedish international Victor Lindelof.
Sky Sports claims that Chelsea are unlikely to allow Matic to join another Premier League club, if the Blues decide to sanction his sale. The English champions would prefer to sanction the midfielder's sale to a foreign club.