The Reds are reportedly hoping to clinch a deal for the 37-year-old from Aston Villa to act as backup for Simon Mignolet, ahead of the current second choice keeper, Brad Jones.
Given had been linked with a move to Doncaster earlier this summer but that switch failed to materialise and he has remained at Aston Villa, even though he has been told by Paul Lambert he has no future at the club.
The 37-year-old has three years left on his £60,000-a-week deal, and Villa are keen to get him off their wage bill.