Ronaldo was fined by Madrid, after attempting to attend Chelsea's home game against Liverpool, back on 7th January.
However, money appears to be no object to Abramovich and he is believed to have made Ronaldo his number one transfer target for the summer.
Ronaldo's agent Rodrigo Paiva has moved to play down talk of a deal being offered, saying nothing will be decided until the summer.
"Ronaldo will not say anything about this at the moment," said Paiva. "He will only comment at the end of the season.
"He is fully concentrated on Real Madrid and is under contract until 2006.
"We have to wait, it is not worth answering these kind of questions at the moment."
Ronaldo was in London during January undertaking commercial work, which prompted talk of a deal being struck with Abramovich, and it seems that the rumours of a move to Chelsea are refusing to go away.