United have spent in excess of £150m on new signings this summer as Mourinho looks to improve upon their sixth-placed finish from last season.
Romelu Lukaku, Victor Lindelof and Nemanja Matic have all joined Mourinho's revolution at Old Trafford.
United haven't won the title since Sir Alex Ferguson announced his retirement from management in 2013.
Since then David Moyes and Louis van Gaal have tried and failed to deliver a top flight trophy.
But Arsenal legend Wright believes Mourinho, with the money he has spent this summer, must deliver by winning the league.
"When you're Manchester United manager that's one of the prerequisites to managing a club of that stature," Wright told Sky Sports.
"The Man United fans and the board have been magnificent with him in respect of supporting him.
"I think Jose, with the money he's spent at Manchester United in his second season which is usually a good one for him, I believe he has to win it."
Mourinho has a fantastic record of winning the top flight in his second season at every club he has managed.
But the United boss is keen to bolster his squad with a winger his main target for the upcoming season.
