Real president Ramon Calderon is a long-term admirer of the young Spanish midfielder.
And on Monday new Bernabeu coach Bernd Schuster confirmed he would love to take the former Barcelona prodigy to the Spanish capital.
Losing Fabregas now would be a disaster for the Gunners, with talisman Thierry Henry already having left for the Nou Camp.
But the midfielder himself told The Daily Star he is in no rush to leave the Emirates Stadium.
He said: "Everyone wants the best for me but I'm happy in London and I haven't made any plans to go.
"Seeing people of all ages, young and old, wearing the shirt with my name on it in London is an honour and gives me a lift. Last season I was the Arsenal player who sold most shirts and that makes me proud.
"I could play every game, the fans were on my side and that means you're doing things well."