Lovren has submitted a formal written transfer request to Saints in a bid to try and force a move away from St Mary’s this summer.
Liverpool are known admirers of the centre half, who last summer became Saints’ most expensive defender when he joined from Lyon for £8.5m, while there is also said to be significant interest from abroad, as well from Tottenham.
Saints have delayed any talks over the future of Lovren or any of their other want away stars until after the World Cup.
They will require that all of them sit down with new boss Ronald Koeman before their futures are decided.
That could happen sooner than Lovren would have hoped after Croatia were dumped out of the World Cup last night.
Lovren, who played the full 90 minutes in all three of Croatia’s games in Brazil, and his teammates needed to beat Mexico in their final Group A game in order to qualify for the knockout stages but they struggled to create a single chance and Mexico were good value for their 3-1 win.
Nearly all the major chances of the match were created by the South Americans.
Hector Herrera struck the bar after 16 minutes while Oribe Peralta produced an awful finish when played through on goal.
In the second half Vedrun Corluka headed a corner off the line before Mexico had a superb penalty appeal for handball controversially waved away.
Mexico took control of the game as Croatia faded still further and scored two goals in the space of three minutes as they bagged a runners-up spot in the group to Brazil and a game against the Netherlands in the last 16.
Rafael Marquez, captaining his side in a record fourth World Cup, headed home the opener from a corner on 72 minutes before Andres Guardado added another three minutes later.
Javier Hernandez secured the result with Mexico’s third eight minutes from time.
Croatia netted a late consolation through Ivan Perisic before their misery was completed when Ante Rebic picked up a straight red card for a horrible challenge.
It left Lovren joining Saints teammates Adam Lallana and Luke Shaw in being knocked out in the group stages and looking to sort out his Saints future in the days and weeks ahead.