Quoting reports in Russian newspaper Izvestia, the article claims the Anfield club are in negotiations with the Russian side with a view to a potential transfer, although any move would likely be put on hold until the end of the season.
The Ivory Coast striker impressed in Anzhi's 1-0 win over Liverpool last week - the 22-year-old scoring the only goal of the game.
And Reds boss Brendan Rodgers is reportedly prepared to sanction a move for the striker, at least according to the player's agent. Traore's representative Dmitry Selyuk told Izvestia that Liverpool are leading the chase for the young forward, although revealed they are not the only interested party.
"Aside from Liverpool, Udinese are another on the list of clubs to have made contact about the purchase of Traore," said Selyuk.
"I'm afraid the Italian side wont be able to meet the player's valuation."
And the player's agent confirmed his client is determined to play in England one day.
"Of course, Russia is not Traore's final resting place," said Selyuk. "The day will come when he will play in the Premier League."
The giant striker only moved to big-spending Anzhi this summer, and currently partners former-Inter Milan striker Samuel Eto'o up front for the Russian club.
Traore moved to Russia last year, joining Kuban Krasnodar from Romanian outfit Cluj, and immediately settled into life in the Russian Premier League, hitting 18 goals in his one and only season with Kuban.
His good form earned a move to Anzhi, and the 22-year-old has adapted impressively, scoring seven goals in 11 Premier League appearances so far for his new club.
But it has been his performances in the Europa League which have caught the attention of Liverpool, who have been reduced to one fit senior striker following injury to Fabio Borini.
Traore is taller than former-Liverpool striker Peter Crouch (6ft 7in), Fulham defender Brede Hangeland (6ft 6in) and Arsenal centre-back Per Mertesacker (6ft 6in). Only Manchester City back-up goalkeeper Costel Pantilimon (6ft 8in) could rival Traore in terms of height should the Ivorian move to the Premier League.