The 37-year-old is said to want to play in the Champions League for the remainder of his career, which the Red Devils won't be doing next season, after finishing sixth.
The Times claims that Premier League rivals Chelsea and Bayern Munich have already discussed the possibility of signing the superstar forward.
New Blues owner Todd Boehly is said to have held talks with Ronaldo's representatives last week.
Bayern's interest is also well-documented, as they look for a replacement for Robert Lewandowski, who looks set for a move to Barcelona.
Ronaldo proved he is still capable of playing at the top level last season, finishing as the Red Devils' top scorer with 24 goals in 38 games.
Reacting to the news on talkSPORT, ex-Manchester United midfielder Darren Gibson, who played alongside Ronaldo during his first Old Trafford spell, expressed his surprise.
He said: "I'm a bit surprised. A lot of people will say it's a good thing but personally I think it would be a bad thing if he left.
"It's up to him if that's something he wants to do: keep breaking records and achieving things."
New Man United manager Erik ten Hag has insisted he plans to utilise Ronaldo in the coming season, while club and international teammate Bruno Fernandes also said he expects him to stay.
"I spoke to him a few weeks ago," Fernandes said. "We're both on vacation, no one bothers friends on vacation.
"I hope to meet him on [July 4] to train there at Manchester United, I'm not expecting more than that.
"There is nothing more than that, but each one takes care of their future.
"I don't believe that the club is willing to lose a player of the value of Cristiano."
After rivals Liverpool and Man City both strengthened, Ronaldo reportedly fears another trophyless season at Old Trafford.
Portuguese outlet Record also claims that Ronaldo 'fears Ten Hag doesn't have weapons to fight for titles', and 'doesn't want to waste' the final years of his career at a club where he feels he can't win trophies.
It could be a very eventful transfer window for Man United, who have made an opening offer of £39million for Ajax defender Lisandro Martinez and continue to pursue Barcelona's Frenkie de Jong.