The Brazilian has enjoyed a fruitful return to the Premier League this season, registering ten goal contributions in 26 league matches after joining on a free transfer.
His one-year contract expires next month, but the 34-year-old is hopeful the club will offer an extension having enjoyed his connection with the Cottagers supporters as much as he did at Chelsea.
The winger said: “I have a contract to the end of June. I don’t know what’s going to happen. I have to speak with the club, and let’s see what happens.
“I feel the same affection for the Fulham fans [as at Chelsea]. If I can stay here, it would be good.
“I am proud of what I have been doing this season. I think a lot of people didn’t believe I could come back to the Premier League and do the same job I was doing a lot of years ago. I am proud of it.”
Willian, who is believed to earn around £40,00-a-week at Fulham, added: “When I left Chelsea, Arsenal didn’t work out as I expected. I went back to Brazil, and some people thought I was finished.
“I’ve proved, not for them but for myself, that I can still play in the Premier League.”
Saturday’s 2-2 draw with Crystal Palace ensured Fulham can reach a new record Premier League points total of 55 for the season if they beat Manchester United at Old Trafford in their final fixture on Sunday.
“We have to be proud of what we did this season,” he said.
“It was an amazing season for us. We have the last game to play and we still have to believe that we can break the record. We have to prepare for the last game.”