Reports in recent days have suggested that van Dijk has offers from a number of Premier League clubs, with City and Chelsea able to offer higher wages and guaranteed Champions League football, with the Reds facing a qualifying round to reach the group stages.
Liverpool are confident they can convince the Dutch international - rated at approximately £50 million by the Saints - to choose them over more lucrative alternatives.
The 25-year-old and his representatives are said to have held positive talks with Liverpool, who are keen to make him one of the club's highest earners and smash their current £35 million transfer record to land him.
Some reports have also stated that playing under Klopp is an appealing prospect for van Dijk, whose main priority is not thought to be his wages.
Liverpool give up hope of Sessegnon deal
But the interest from various clubs in the former Celtic man means the transfer could prove complicated, with the Reds having recently conceded defeat in the pursuit of Fulham left-back Ryan Sessegnon.
It has been reported for several months that Liverpool viewed the 17-year-old as their first-choice option to add competition for, and to eventually replace, James Milner.
But the teenager is now thought to be nearing a move to Tottenham Hotspur amidst suggestions that he wasn't keen on moving far from his London home, with Liverpool having conducted unsuccessful talks with Sessegnon and his agent.
The England U19 international, who scored seven goals in 30 appearances for the Championship side last term, but Liverpool have given up any hope of attracting him to Merseyside with Spurs now the favourites to sign him.
Liverpool will now look to other targets, with Alberto Moreno expected to leave as part of a loan deal or on a permanent basis and only Milner left as a senior option at left-back.
The Merseyside outfit have already made their first signing of the summer with young Chelsea frontman Dominic Solanke joining in a £3 million deal and going into the under-23s team.