The 28-year-old will have one year remaining on his deal this summer but is happy to remain at the Etihad Stadium if he continues to play a part in Pellegrini's squad rotation.
The City boss is an admirer of Milner's versatility and the club are keen to complete a deal before the player sets off for England's World Cup campaign in Brazil.
City would be willing to offer Milner a three-year extension on his current £85,000-a-week salary.