The Reds have been heavily linked with the Saints’ silky playmaker, with a £20m deal expected to be completed soon, as the player himself is said to be desperate to make the switch.
Now Brendan Rodgers is keen to pay another £20m to snap up Lovren in the same deal.
The Croatian international has enjoyed a fantastic debut season in the Premier League, helping the South Coast outfit to secure a stable midtable finish.
The powerful defender is seen as the ideal man to impose some authority at Anfield, with the Merseysiders having shipped a whopping 50 goals despite finishing runners-up to Manchester City in the title race.
Rodgers himself is thought to be unconvinced by vice-captain Daniel Agger, which means he could sold in a bid to part-fund a move for Lovren.
Lallana remains the top target, with the composed passer seen as the perfect player to rotate with Philippe Coutinho and Raheem Sterling in an attacking midfield set-up.