De Vrij is due to fall out of contract in the summer, and could walk away on a free transfer unless the Rome-based outfit can convince him to pen a new one.
Talks over a new deal are reported to have stalled, and unless Lazio can reignite a deal quickly, they may have to consider the option of selling when the transfer window re-opens.
De Vrij will not be short of offers either, and does have the option of staying in Serie A if he so wishes. Both Italian champions Juventus and Inter Milan are known to be keen and will make moves of their own in the new year.
However Liverpool could offer him a fresh challenge in the Premier League and he would cost considerably less than Virgil van Dijk, who still remains a Southampton player.
A goal apiece from James Tomkins and Luka Milivojevic handed Crystal Palace a crucial 2-0 win over Huddersfield at the John Smith's Stadium on Saturday evening.
Manchester United bounced back from their Champions League exit as they beat Brighton 2-0 at Old Trafford on Saturday night to reach the FA Cup semi-finals.
Tottenham reached the FA Cup semi-finals after they cruised to a 3-0 victory over Swansea in their quarter-final clash at the Liberty Stadium on Saturday.