Although United have tried to convince Rooney to stay following Sir Alex Ferguson's retirement, the 27-year-old is adamant that he wants to leave.
The problems stretch back over the past year under Ferguson, with Rooney unhappy at how often he was played out of position.
He has also been hurt by some of the suggestions that appear to have emanated from Old Trafford relating to his lifestyle and fitness.
While Chelsea lead the chase for Rooney, Arsenal should not be ruled out. Though most attracted by a move to Stamford Bridge, he would give their London rivals serious consideration and it is conceivable that United would rather do business with Arsene Wenger than Mourinho.
Along with Manchester City, Chelsea are regarded as United's principal rivals for the English Premier League title and it could be considered less risky to sell Rooney to Arsenal.
It is understood that the Rooney situation is beginning to cause tensions in the dressing room, but United remain adamant that they will not sell at any price.
United insist that they will not soften their stance over Rooney but the saga is threatening to damage their pre-season.
Rooney returned early from the club's summer tour with a hamstring strain and has not played since the end of last season.
His lack of match practice makes him a major doubt for the Community Shield on Sunday against Wigan Athletic, for England's game against Scotland at Wembley on Wednesday week and, increasingly, for the start of the Premier League.
Rooney's chances of being allowed to leave United have hardly been helped by the club's failure to land any of their main transfer targets. They are still trying to sign Cesc Fabregas from Barcelona and are preparing a third bid of £35 million.
Wenger, however, has already made inquiries on behalf of Arsenal, who have an option to buy back the midfielder.
Arsenal have also made an inquiry for Rooney, although their primary target is still Luis Suarez. Liverpool rejected Arsenal's unusual offer of £40,000,001 but the forward is still keen to leave Anfield.
Arsenal are hoping that Suarez will push to leave in the belief that a clause in his contract allows him to talk to another club if Liverpool are not in the Champions League and receive an offer of more than £40 million.