The 34-year-old has been training with Fulham after terminating his contract with Brazilian side Corinthians.
His family want to return to Europe after he received abuse on social media in his native Brazil.
Willian has been living out of a hotel for the last fortnight and would welcome a return to London.
Fulham are not under pressure to rush through a deal for Willian, who could sign after the transfer window shuts because he is a free agent.
Manager Marco Silva is keen to add more numbers and quality to his squad before Thursday’s deadline.
Fulham are in the market for a left-sided defender, with Paris Saint-Germain’s Layvin Kurzawa discussed, and have reopened talks with Roma over forward Justin Kluivert.
Silva said ahead of the transfer deadline: “It is so clear for everyone. The board has known from last April, not from last week or the last 15 days or the last month, what the plan was and what we should do in the market.
“Unfortunately, they are coming really late for us because we’ve already played four games at this level.
“It is not important to say how many or the positions. I think everyone can see and analyse our team. They can see what the positions are, and that we need to add some numbers.”