The United States international is desperate to leave Craven Cottage this summer and pulled out of the clubs pre-season training camp in Switzerland after holding talks with manager Martin Jol.
Dempsey wants to play Champions League football having served the Cottagers for five-years and has been heavily linked with a move to Anfield throughout the summer.
Reds chief Brendan Rodgers has already confirmed his intentions to bring the 29-year-old to Merseyside revealing a fortnight ago that they’ve already made an approach.
“Clint is a player we’ve inquired about, it is as simple as that,” Rodgers told the Guardian.
“Ian Ayre, our managing director, has spoken with the club to see what the position is. That is where we’re at. He’s a very talented player but we don’t like to talk about other clubs’ players.”
It’s now emerged that a deal to sign Dempsey, who scored 23 goals in 45 games for Fulham last term, is nearing completion and will become the Northern Irishman’s second signing since taking over from Kenny Dalglish in June.
Reports indicate that Liverpool will pay significantly less than the £10 million fee demanded by Jol with the attacker entering the last year of his contract with £6m thought to be enough to prise him away from the Thames.
Stewart Downing and Charlie Adam are being mooted as makeweights to smooth over relations with the West Londoners after both failed to impress their new boss.
Rodgers is also looking to sell club record signing Andy Carroll before the new campaign gets underway with West Ham United having a £2 million loan bid accepted on Monday.
Hammers boss Sam Allardyce managed the 23-year-old England international during his short spell at Newcastle United but it’s thought Carroll is angling for a move back to St James’ Park.