The Juventus midfielder has been heavily linked with a return to his former club but a deal is still no closer to completion despite widespread reports that his transfer was imminent.
For the second time this week Mourinho denied that Pogba was the missing piece in his jigsaw, but hinted there was one week left for United to wrap up the signing before their Premier League campaign gets underway on August 14 with a trip to Bournemouth.
Speaking at a press conference ahead of his side's Community Shield clash with Leicester on Sunday, the Portuguese coach said: "I don't like to speak about other players. Other managers like to do that.
"What matters to me is only what happens in my house, not the house of my neighbour. Pogba is a Juventus player until officially he isn't. It stays like this.
"Honestly, the market closes on August 31, but I think my club is trying to do everything to close our market before August 14. So hopefully by next week we will have our squad closed."
Mourinho admitted to varying levels of fitness from his players in their final week of pre-season preparations, but believes they are "ready to compete" against the Cherries next weekend.
"One thing is to start five weeks ago, like some of them, and another thing is to start one week ago - it is completely different," he declared.
"But the way we work we try to bring them on as fast as we can. I won't say to top form but to a level where they are ready to compete. That is the reality of modern football."