Rooney, 31, has been sidelined by manager Jose Mourinho heightening expectations he could leave United for the mega-rich Chinese Super League.
Guangzhou Evergrande, Beijing Guoan, Jiangsu Suning and Tianjin Quanjian are said to be among those interested in United's all-time top scorer.
British media have reported that he could earn up to $1 million a week, making him the world's best paid player.
However it remains unlikely that any move would be complete before the transfer window shuts next Tuesday.
Rooney missed Wednesday's Europa League last-32 second leg in Saint-Etienne through injury, but dominated conversation in France.
Mourinho said that Rooney's future was in his own hands in comments at his pre-match press conference that only fuelled the speculation.
"You'll have to ask him," Mourinho said. "You have to ask him.
"Of course I can't guarantee (he will be here). I can't guarantee that I'm here next week, how can I guarantee that a player is here next season?
Pushed on whether it was possible Rooney could leave in the next week, Mourinho retorted: "You have to ask him, not me.
"I was very open with you in the answer. I don't want him to leave."
Rooney's deal at Old Trafford runs until mid-2018, with United holding the option to extend it by a further year.
He is United's record goalscorer and has won five Premier League titles and a Champions League trophy since joining as an 18-year-old for £27m from Everton in 2004.
A return to boyhood club Everton and move to Major League Soccer could be other avenues for Rooney, but are unlikely to offer wages anywhere near that available in China.
Former United team-mate Carlos Tevez, ex-Chelsea midfielder Oscar and former Watford striker Odion Ighalo are among those to have recently made big-money moves to the Far East.
However, a switch to China could bring Rooney's England career to an abrupt end.
The country's all-time top scorer has won 119 caps and is closing in on Peter Shilton's record of 125 appearances.
Rooney has already announced plans to retire from international football after the next year's World Cup in Russia.