The Sunday People says Potters manager Mark Hughes will try to tempt Rooney to ignore offers from China and the United States and stay in the Premier League with Stoke.
The report claims Stoke can fund the move thanks to their sponsors Bet365 – the online gambling company, founded by the club's chairman Peter Coates.
However, the Potters could find it difficult to persuade Rooney to come to the Britannia Stadium, with the 31-year-old having made it clear that he would not play for any team in the Premier League other than United and boyhood club Everton.
Rooney has scored five goals in 25 league appearances for United this season. He has two years left on his Red Devils contract, worth a reported £300,000 per week.
Speaking after United's Europa League final victory over Ajax in Stockholm on 24 May, Rooney said he had "more or less" made a decision on his future and added that he had a number of offers from England and abroad.
"I've got decisions to make now over the next few weeks, have a word with my family, and then I'll decide. I think I just have to make a decision in terms of a football decision, and that's what I'll do," he was quoted as saying by Sky Sports.
"I'll do that the next few weeks with my family while I'm away and, honestly, as soon as I know what's going on then, I'm sure you [will know]."
United boss Jose Mourinho described Rooney as a "very important player" and said he would be happy for the striker to stay at Old Trafford next season.
The 31-year-old has been linked with a big-money move to China, with his agent Paul Stretford flying to the country to hold talks with Chinese Super League clubs in February.