The Reds were keen on landing the Dutch international earlier in June. However, Liverpool were forced to end their interest and issue a public apology for "any misunderstanding" regarding the central defender. They were accused of an illegal approach.
According to the Mirror, Van Dijk is still keen on joining Jurgen Klopp's side and is close to forcing a way out of the south coast club. Southampton manager Mauricio Pellegrino left his captain out of the squad for the pre-season trip to France as the defender was psychologically not prepared to join the rest of the teammates.
The former Celtic man has already communicated his decision to his current employers that he will not play for the Saints. He has set his sights on leaving the club and make a switch to Anfield.
Southampton are unwilling to do business with Liverpool and their executive director Les Reed was determined not to sanction the Dutchman's sale to the Reds. However, the 26-year-old's desire to leave the club has put them in a difficult situation.
Pellegrino is left frustrated as he believes Van Dijk's saga will have an impact on the team's morale, with the new season set to kick start in less than two weeks time. Chelsea and Manchester City were also looking at the option of signing the centre-back.
However, the Premier League winners and Pep Guardiola's side were made aware that Van Dijk is only looking for a move to Liverpool in the ongoing transfer window. This forced to end their interest and the 18-time English champions could make an approach in signing the player.
The Merseyside club are ready to pay close to £60m ($79.4m) for Southampton star, who signed a six-year deal at St Mary's Stadium last year. Should Reed back down and sanction Van Dijk's sale to Liverpool, he will be Klopp's fourth signing of the summer. The Reds have already completed deals for Dominic Solanke, Mohamed Salah and Andrew Robertson.