The Liverpool Echo claims rumours that Van Dijk has decided his future are untrue and that the Reds have been "encouraged" by feedback they have received from the centre-back's representatives.
However, the Echo said Van Dijk would not make a decision on his future solely based on money and that he had been impressed by manager Jurgen Klopp's vision at Liverpool.
The 25-year-old is the Reds' priority summer transfer target, with the club willing go above the record £35m they paid to Newcastle United for Andy Carroll in 2011 for his services.
Chelsea have also been linked with a move for the Netherlands international.
Southampton are reported to be holding out for a transfer fee of £60m for Van Dijk as they are obligated to pay 10% of any fee they get for the defender to Celtic. The Saints signed the 25-year-old from the Scottish club for £13m in 2015.
Southampton boss Claude Puel has insisted that Van Dijk would stay at St Mary's next season, pointing out that he was contracted to the club until 2022.
"He's our captain and he's important for us. He's matured and taken leadership on the team; he's a fantastic player we know well," Puel was quoted as saying by the Daily Telegraph in May.
"He's important for us and now, of course, we will wait to the beginning of the season to see him on the pitch with Southampton. He has a long contract and the directors will want to keep him."