Cole rejected a new contract at Stamford Bridge this week, according to The Sun.
Real Madrid coach Manuel Pellegrini has been a long-term admirer of the former Arsenal full-back.
A source close to Cole is quoted by The Sun as saying: “Ashley loves Chelsea and is genuinely excited about working with new boss Carlo Ancelotti.
“He wants to give the best years of his career to the club but the deal has to be right for such an important contract.
“He needs to feel as valued as any of the club’s other top players.”