The 34-year-old winger is evaluating the latest offer from the Cottagers after turning down an initial one-year contract last week.
Willian is technically now a free agent and will look for an alternative Premier League move if he chooses to leave Fulham.
Marco Silva is also keen to tie down his defence with left-back Antonee Robinson increasingly tipped to sign a new deal at Craven Cottage.
Tosin Adarabioyo has turned down a contract with just one year left on his deal and Fulham are keen to prioritise signing a replacement.
They had a £10million bid turned down by Benfica for 22-year-old centre-back Morato on Wednesday, while Fiorentina have snubbed an informal £8m offer for Igor.
Benfica have tried to ward off further advances by demanding £40m for Morato but Fiorentina are more open to doing a deal, asking for around £20m.
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