Out-going boss Sir Alex Ferguson confirmed after his last home game against Swansea earlier this month that Rooney had handed in a transfer request - the second of his Manchester United career.
Moyes, who was in charge of Everton when Rooney was sold to the Red Devils in 2004, now has the difficult task of handling the England international and either letting him leave for the right price or finding a place to keep him happy within his new squad.
Rooney, who pulled a similar stunt in 2010 while questioning Manchester United's ambition, has found himself usurped by the arrival of Robin van Persie last summer which has seen the 27-year-old played in various other positions and left out of key games.
However the Guardian report that Moyes, who visited United's Carrington training this week ahead of his first day on July 1, is confident of persuading Rooney that the best place for him is in Manchester.
The paper claim that since Ferguson announced his departure and Moyes was named named his successor Rooney is open to wiping the slate clean and looking to continue his Manchester United career.
The key to Rooney's future will come when contract negotiations open this summer. The former Everton man has two years left on his current contract but United are unlikely to be willing to offer a pay-rise from his current wages of around £300,000-per-week.
While Rooney's future remains in the air, one man who looks like staying put at Old Trafford is Rio Ferdinand. The defender is set to sign a new contract extension with the club when he returns from holiday.
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