Ronaldo, 37, is said to want to play Champions League football again next season.
The Portugal striker was United’s top scorer last season with 24 goals - 18 of those coming in the Premier League, which saw him finish as the third top scorer in the division- but the Red Devils could only finish sixth in the table.
Ronaldo has been part of 19 consecutive Champions League seasons since leaving Sporting Lisbon for United in 2003.
The forward has made 183 appearances in the competition, scoring 140 goals and adding a further 42 assists.
Ronaldo believes he still has "three or four years" left at the highest level of the game, The Times reported.
Ronaldo returned to Old Trafford last summer from Juventus and still has a year left on his current deal.
Chelsea, Bayern Munich and Napoli are among the clubs said to be monitoring the situation.
United will be hoping to the recoup the £12.9 million they paid out last summer to bring him back to Old Trafford.