The Tottenham defender sparked speculation over his future earlier this week by criticising the club's lack of signings in a newspaper interview.
Rose is not demanding a move away from Spurs but is reportedly open to joining United or Chelsea if they decide to sell.
BetStars Director of Trading said Ian Marmion said: "Danny Rose may have publicly apologised for the comments he made earlier this week criticising Spurs manager Mauricio Pochettino and chairman Daniel Levy, but the odds suggest that the damage has already been done, with Manchester United leading the chase for the England international.
"The Red Devils are the clear favourites to sign Rose at 11/8, with Premier League rivals Chelsea currently priced at 4/1 and Manchester City at 5/1.
"The lure of playing at Old Trafford could be too much to resist for Rose, so we’re calling it that he’ll be wearing a Manchester United shirt before the transfer window closes."
In his interview with The Sun, Rose also said that he believed he and his team-mates were underpaid but he has since apologised.
In a statement released through his agent, the left-back said: "Having had time to consider my actions, I acknowledge now that the timing and manner of what I said was ill-judged.
"My words were not meant to offend and for that I would like to apologise to the chairman, manager, my team-mates and the fans.
"I'd also like to wish the lads good luck ahead of Sunday's game against Newcastle."
