The 33-year-old, whose current deal expires at the end of the season, is believed to be holding out for a two-year contract from the Spanish giants.
With Los Blancos reportedly tabling a 12-month extension, Pepe's advisors are open to looking elsewhere, and it is rumoured that the Portugal international would consider a switch to Chelsea.
Despite the speculation, Pepe has suggested that he still wants to remain at Madrid, where he joined from Porto in 2007.
"I'm relaxed about it, I want to help Real Madrid, a club which I love a lot," Pepe told Spanish TV station Antena 3. "My time will come. I'm calm.
"I hope that the club calls me first to resolve the situation. I'm relaxed, the people at the club know that this is my home. As I have said before, I will wait until the last second for Madrid."
Should the two parties fail to agree fresh terms by January 1, Pepe will be free to speak to other clubs regarding a transfer.