The Reds have been heavily linked with the Croatian World Cup star in recent weeks amid talk that he requested to leave the South Coast club before jetting off to Brazil.
Lovren was one of the stars of the Saints’ impressive 2013/14 campaign, with is robust displays helping them to an eighth placed finish.
But Liverpool are confident that a £15m offer will be enough to lure the 24-year-old away from St Mary’s Stadium just 12 months after his arrival from Lyon.
The centre-back himself is understood to have an eye on Champions League action, which makes keeping him at Southampton a difficult task for new manager Ronald Koeman.
To make matters worse other stars such as Morgan Schneiderlin are also being linked with switches away from the club.
Meanwhile, Liverpool are looking at Chelsea left-back Ryan Bertrand after growing frustrated in their pursuit of Alberto Moreno.
Sevilla want £20m for the Spaniard, making the Blues outcast – who spent the end of last season on loan at Aston Villa – a more cost-effective option for Brendan Rodgers.