While the Red Devils did win the Europa League and EFL Cup last season, they ultimately achieved a disappointing 6th place finish in Mourinho's debut season, 24 points adrift of champions Chelsea.
Since then, United have splurged in the transfer market, spending nearly £150m ($194.8m) on the likes of Romelu Lukaku, Nemanja Matic and Victor Lindelof as they look to improve on their league table finish last season.
Keeping the level of summer investment in mind as well as Mourinho's knack of doing well in his second seasons, Wright claims the Portuguese manager has to deliver the title to Old Trafford after what has been four years without league glory for United supporters.
"When you're United manager it's [the title] a pre-requisite to managing a club of that stature," Wright told Sky Sports, as quoted on The Express.
"United fans and the board have been magnificent to him in terms of supporting him."
The former Gunners striker also says expectations on the other side of Manchester will be high as well but ultimately believes that Mourinho has to win it.
"I believe that them and City have to do well, as [Pep] Guardiola is under an amazing amount of pressure," he added. "The amount Jose has spent, and the fact his second season at a club is normally a good one, then he has to win it."
United most recently lost 2-1 to Real Madrid in the European Super Cup on Tuesday (8 August) but showed slight signs of promise.
They kick off their league campaign against West Ham United on Sunday (13 August).