The Croatian international only joined the Saints last summer, but his terrific displays across last season have prompted links with Liverpool.
The Reds are rumoured to be interested in the centre-back after moving to secure Rickie Lambert and making contact over a deal for captain Adam Lallana.
It is understood that Lovren submitted a request to leave St Mary’s Stadium at the very end of last month, citing managerial turmoil – Maurico Pochettino has left to join Tottenham – and a desire to play Champions League football – further hinting at a move to Anfield – as major factors in his decision.
Despite the player’s desire, the club are yet to make an official response, and have maintained that no more star names will be allowed to leave the club until a new head coach is in place.
The 24-year-old is said to be one of Liverpool’s top targets, and they may be able to negotiate a reduced fee with the South Coast outfit now that Lovren’s desires are known.
However they are not the only side interested, with Spurs also in the mix.
Lallana is also keen to move to Merseyside, while teenage left-back Luke Shaw is said to be close to joining Manchester United.