The People says the Scot has made keeping Rooney his top priority - and could even meet the player this week to settle any grudges the England striker had with predecessor Sir Alex Ferguson.
There is now a growing acceptance at the club that Rooney did not ask for a transfer when he met Ferguson last month.
Ferguson twice repeated his claim that Rooney wanted out.
But the player's advisers have insisted the striker was only voicing his concerns about the way he was being deployed.
Moyes has now made it clear he is excited by the prospect of linking up with the player to whom he handed an Everton debut at the age of 16.
With two years left on Rooney's £250,000-a-week contract, much will still depend on whether an extended deal can be agreed.