The Mail on Sunday says Rooney had become disillusioned after believing that he was being marginalised by Sir Alex Ferguson and asked for a transfer last month, having become resigned to the thought that his prospects at United were limited.
The striker is understood to be keen to start afresh under Moyes, his former manager at Everton, and is seeking reassurance about his future.
The pair once fell out over false claims made in the player's autobiography about Moyes leaking information to the press.
Since then, though, Moyes has spoken warmly of Rooney after he said the player phoned him to apologise personally and to thank his old boss for the guidance he gave early in his career.
Moyes is expected to meet Rooney, before taking a holiday at the end of this month, as one of the more pressing issues at the start of his Old Trafford reign.