The highly-rated 25-year-old only moved to Anzhi from Shakhtar Donetsk in January but the Russian side are now looking to offload their star players, because billionaire owner Suleiman Kerimov is scaling back his financial commitment to the club.
And Liverpool are eager to strengthen their attacking options, having failed to land Henrikh Mkhitaryan, who joined Borussia Dortmund from Shakhtar, and Atletico Madrid's Diego Costa.
Willian recently told ESPN Brazil: "He (Kerimov) made it clear that, if good bids arrive, he would certainly help us go.
"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."
Liverpool may face competition for the signature of Willian, who has come close to joining Tottenham and Manchester City in the past.
But the Reds now seem to have the inside track on signing the former Corinthians man, who would be likely to cost in the region of £30m.