The Daily Star says Moyes does not take over from Sir Alex Ferguson until July 1 but, like United, he is keen to address the Rooney problem as soon as possible.
It is believed that Moyes, who managed Rooney at Everton, wants his former player to stay - despite a legal issue between them in the past.
That was resolved and now Moyes has another Roo issue to tackle.
Rooney could be persuaded to stay at Old Trafford if he likes what he hears from Ferguson's successor about his role next season.
United, who have stated that Rooney is not for sale, would like the situation resolved this week before the striker links up with the England squad on Sunday.