The Daily Mail say the Real Madrid playmaker has his eye on west London rather than the Theatre of Dreams.
Both sides have made contact with the Colombian recently, but it’s the Blues that are thought to be winning the race.
“I have offers and I’ve got seven days to think about it”
Real are now resigned to losing Rodriguez after his comments at the weekend.
He said: “I’m happy here, but I want to play more.
“I want to stay at Madrid forever, but you have to think about everything and look at your options.
“I have offers and I’ve got seven days to think about it.”
Madrid have been offering his services around after their transfer ban was reduced by a Court of Arbitration in Sport ruling which overturned FIFA’s decision to block them from doing deals for 12 months.
Earlier today reports in Colombia said the playmaker had visited the British embassy in the country in a bid to get a visa application moving.