The Southampton captain has established himself as one of the finest central defenders in the Premier League since moving from Celtic for £11.5m in September 2015.
Liverpool are refusing to comment on the speculation, but it is understood they are prepared to break their transfer record to land him.
Personal terms have not been finalised with Van Dijk, but Liverpool are prepared to match the contract offer from Manchester City which could be as much as £200,000 a week.
Pep Guardiola has identified him as a priority target as he looks to strengthen his defensive options this summer, though City are concerned about the price they may be forced to pay to land him.