The Reds showed an interest in the striker during the winter transfer window and the Daily Mail claim the Anfield side are willing to do business again to secure his signature.
However, Liverpool are less keen on the terms being offered with Harry Redknapp demanding Benayoun plus an additional £2m heading into their coffers.
The Merseysiders could find better value in doing a straight cash deal for their Israel midfielder and would hope to raise as much as £10m by exploring other interest from Aston Villa, Roma, Sevilla and Dynamo Moscow.
Meanwhile, Lokomotiv Moscow fear that their hopes of signing Pavlyuchenko from Tottenham Hotspur are dead after the striker's end of season return to starting line-up at White Hart Lane.
The Russians almost signed him in January for £10m but the goalposts have moved since then according to head coach Yuri Semin.
He said: "The situation has changed. I'm afraid it will be difficult to get Pavlyuchenko. His team qualified for the Champions League and he regained the coach's confidence."