The Red Devils completed their third signing of the summer on Monday when they tied up a deal to bring midfielder Nemanja Matic to the club from Chelsea.
Manchester United have also completed deals to sign Victor Lindelof and Romelu Lukaku this summer as Mourinho prepares his team for his second campaign in charge at the club.
The Portuguese coach is believed to have asked executive vice-chairman Ed Woodward for four new signings this summer.
And although Mourinho admits that he has been happy with how Manchester United have conducted their business so far this summer, he still wants to bring in at least one more player if possible.
Speaking at a news conference in Dublin on Wednesday night, Mourinho said: “I was hoping for four plays to improve the squad.
“The club did a fantastic job by getting three of them which is very difficult with the crazy market where we are now.
“To get three out of four, I thank the club for that. I’ll be happier if they give me four out of four. But they did a great effort for me.”
Manchester United kick off their Premier League campaign with a home clash against West Ham United at Old Trafford later this month.
The Red Devils take on Real Madrid in the Uefa Super Cup in Macedonia on Tuesday as they look to win the first trophy available to them this season.
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