De Vrij’s Lazio career had been plagued by his injury, but he found fitness and form last term as he helped his side finish fifth in the league.
Liverpool were not only the only European club keeping tabs on the defender, who has been capped 31 times by the Netherlands.
The 25-year old was not expected to extend his stay at the Serie A outfit but he is now said to have reconsidered their offer.
The new contract would include a €30million release clause and could be signed next week, which would represent a fresh blow to Jurgen Klopp as he looks to strengthen his back line.
If indeed De Vrij does commit to Lazio, Liverpool could turn to their Serie A rivals Napoli for the in-demand Kalidou Koulibaly, while Bayer Leverkusen’s Jonathan Tah has also been mooted as a potential option and another move for Virgil van Dijk is still on the cards too.
Liverpool are currently short on reliable cover in the defensive department, an issue Klopp will look to address as he prepares his side for a return to the Champions League.