The Dutchman is due to fall out of contract at the end of the season, but Lazio are keen to get him tied down to a new one that will include a 45million euro buy-out clause.
However, Di Vrij will only sign if the clause is set at 25million euros and unless a deal can be agreed, the central defender will be heading for the exit door either in January or at the end of the season.
Liverpool are said to be watching developments closely and could be ready to make an offer in January’s transfer window, when the player will be free to talk to other clubs.
Fellow Dutchman Virgil van Dijk remains Jurgen Klopp’s number one target but the Reds are set to look elsewhere after failing to sign the Southampton ace in the summer, with Klopp weighing up alternatives.
Preston are reportedly weighing up whether or not to make a fourth offer for Peterborough United left-back Andrew Hughes.