A stunning campaign that has seen the American score 22 goals in all competitions for the Cottagers has alerted a number of teams with the Reds leading the chase for his signature.
It’s widely thought the 29-year-old will leave Craven Cottage at the end of the season in search of European football and Dalglish is desperate to bring him to Anfield.
The Kop legend has suffered a difficult first year back in the Liverpool hot seat with the club sliding away from the Premier League top-four and into mid-table.
A potential cup double has eased the pressure building the 61-year-old’s shoulders but owner John W. Henry will expect to see the club challenging back in the top-flights upper echelons next season.
Dalglish wants Dempsey to spearhead his Merseyside revival and is prepared to test Fulham boss Martin Jol’s resistance to keep the forward with a sizeable bid in the summer after being granted funds to spend.
Co-owner Tom Werner recently admitted he would be the ideal signing for Liverpool: “We never talk publicly about our transfer targets, however, any team would be lucky to have Clint Dempsey playing on their squad”
The Reds are expected to face stiff competition for the former New England Revolution man with rumours circulating that Arsenal chief Arsene Wenger is keen to acquire his services.
The Frenchman already has German international striker Lukas Podolski heading of the Emirates Stadium but wants Dempsey to create a devastating attacking trio led by club captain Robin Van Persie.
Should the Gunners win the race Dalglish will turn to Montpellier’s Moroccan international Younes Belhanda who has shone in midfield for the Ligue 1 leaders this season.