Rooney still wants to leave the club despite Sir Alex Ferguson's departure, according to the Daily Mail.
Recent reports suggested Rooney was unhappy at Ferguson's claim he had requested a transfer - suggesting a willingness to stay under new manager David Moyes.
But the Mail reports that Rooney has not been persuaded his future lies at Old Trafford, forcing United to contemplate an asking price for the England striker.
The 27-year-old has two years left on his contract and is expected to cost around £25 million, slightly less than United paid Everton for him in 2004.
Arsenal are very marginal favourites to sign Rooney, but Chelsea, Paris Saint-Germain and Real Madrid are all realistic alternatives.
The Times reports that Moyes still hopes to talk Rooney round and will hold discussions with the player and his agent Paul Stretford next week.
Despite Rooney delivering a disappointing return on his £250,000-a-week salary last season, United still consider him a key player. The club's new chief executive Ed Woodward is keen to make a strong start to both his and Moyes's tenure, and selling arguably the club's most marketable star would send out a negative message.
Meanwhile, the Daily Telegraph says United are preparing a second bid for Leighton Baines after their original £12m bid was rebuffed.
The Guardian reports that a fee in the region of £18m will be necessary to prise him from Goodison Park.
United's interest in Baines appears to have upset the club's current left-back Evra, whom The Sun claims is ready to quit Old Trafford.
The Frenchman is not prepared to stick around and compete for his place in the side if Baines arrives. He has been linked with a return to Monaco, who have already spent over £100m on Radamel Falcao, James Rodriguez and Joao Moutinho in an impressive early-summer spree.
The 31-year-old remains a formidable defensive presence but has suffered badly with injury in recent seasons.
Milan see Vidic as the man to shore up their back line as they attempt to wrest control of Italian football back from Juventus.