Pope has just two years remaining on his contract with the Clarets, and is currently on international duty with the England squad, but in his absence, Jurgen Klopp's side are planning a swoop for the 26-year-old.
Simon Mignolet, Klopp's former first-choice, was dropped in favour of Karius, but after two massive blunders cost Liverpool a shot at Champions League glory, the club are looking elsewhere.
The greatest challenge that they face is finding a suitable replacement at a reasonable price. The Reds had previously been eyeing up Roma's Alisson but his £70million valuation would be enough to put many clubs off.
For his part, Pope already has experience in the Premier League, would come at a far more justifiable cost and Burnley have managed to secure a Europa League qualification spot next season.