With United keen and interest likely to come from the likes of Paris Saint-Germain and clubs from the Chinese Super League, they face a tough task to persuade the Portugal global to move to Stamford Bridge.
The rumours of a possible transfer back to England by the two-time Ballon d'Or victor started after a report surfaced that Spanish authorities have accused the 32-year-old of avoiding £14.7m in taxes.
The former Manchester United winger is angry about the way he has been treated by Spanish tax authorities and has told Real he wants a move away from the club.
The Portugal global shocked the world of football after declaring that he wants to leave Spain amid reports that he defrauded the Spanish taxman for £13million.
According to Italian football expert, Gianluca Di Marzio, Ronaldo has told his agent, Jorge Mendes, to do "everything he can" to make the move a reality.More news: Mexico vs. Portugal: Confederations Cup highlights, score
He has also won two La Liga titles with the club he joined from Manchester United for a then world record fee of 96 million euros ($A140.9 million) in 2009.
"I had the same situation with Manchester United when he wanted to leave and I signed him".
"The club has been very important in his career".
Marca also reported that club captain Sergio Ramos had called Ronaldo to ask him to stay.
"He says I want to leave and that's it, as a president you can't do anything other than let him go and get the best possible deal".