The 34-year-old Brazilian has less than six months left on his deal but increasingly regards west London as home and wants to extend his deal by an extra year.
Willian joined Fulham on a free transfer last summer and has been a key player in their brilliant start to the season under Marco Silva. He has three goals and two assists in 14 Premier League appearances.
Fulham have indicated they are willing to give Willian a contract extension but talks will not start until the transfer window has closed, as the club prioritise new signings to strengthen their squad.
Silva wants to bring in a right-back and Fulham are interested in Heerenveen’s Milan van Ewijk, FC Twente’s Joshua Brenet and Rennes’ Hamari Traore.
Fulham had identified Arsenal’s Cedric Soares as their top target but they are reluctant to pay his £65,000-a-week wages in full a move for the 31-year-old has stalled.