The Russian club is keen to sell some of their assets as billionaire owner Suleyman Kerimov looks to slash costs and recoup the £30million paid to Shakhtar Donetsk for Willian in January.
Liverpool have made a £5million profit in the transfer market so far this summer and have the funds to satisfy Anzhi but Willian must agree to a cut in his £100,000 per week salary for the move to be possible.
"I'm hoping in the coming days to receive an offer from a club like Liverpool - it sure is a great club," Willian said.
"Of course England is a league that many dream of playing in and I'm no different.
"My agent has had conversations with some clubs.
"I love the Premier League and the style of English football and certainly if it happens it will be good for me and I'll grow professionally.
"(Kerimov) doesn't want to let anyone go for free but he made it clear that if good bids arrive, something at least close to what he paid for the players, he would certainly help us go."
A deal would see Willian become Liverpool's second most expensive player in the club's history, behind £35million flop Andy Carroll.
The 25-year-old Brazilian was subject to a rejected £34million bid by Manchester City last month, before Kerimov decided to cut costs and has also attracted interest from Chelsea and Tottenham.
With Liverpool ramping up their dealings in the transfer market, they have also been linked with £12million-rated Granada's Guilherme Siqueira but must compete with Real Madrid for the left-back's signature.
