The Merseysiders hope to reach an agreement with his club Anzhi Makhachkala in the next 48 hours after lodging a formal bid of 30 million euros for the in-demand forward.
Liverpool has indicated that it is willing to match Willian's current €100,000-a-week salary in order to convince him to move to Anfield.
The Reds have also attempted to smooth negotiations with Anzhi by offering the Russian club a sell-on clause in which it would receive 50 percent of the profits of any future sale.
Tottenham has also held talks about buying Willian, who is a long-term target of the club, but is not yet believed to have made an offer.
The 25-year-old is expected to make a decision about his next destination by Tuesday night, with the likelihood growing that a deal could be formalized by the middle of the week.
The Brazilian revealed earlier in August that he was "hopeful" that a move to the Reds could come to pass this summer amid Anzhi being willing to offload some of its biggest earners, including Samuel Eto'o, in an attempt to trim costs.
Willian said: "For now, all there is are talks. There was the Manchester City bid, and that was turned down, and that was a while ago.
"Now, I'm waiting for a new bid to arrive in the next days from a few clubs, like Liverpool. If it's really Liverpool, then it is surely a great club. I'm hopeful and very calm, just waiting for the best."