Holland international De Vrij has been with the Italian outfit since joining from Feyenoord in 2014 and he has been a regular first-team fixture, making 62 Serie A appearances and scoring two goals.
Several top European clubs were said to be showing a keen interest in the centre-back in the recent summer transfer window with Liverpool, who missed out on Southampton's Virgil van Dijk, keeping an eye on his situation.
The Reds may have received a boost in a potential January move with reports claiming talks over extending the 25-year-old's stay at the Stadio Olimpico have now stalled.
With De Vrij's current deal expiring at the end of the campaign, the Biancocelesti are desperate to tie him down to a longer contract as the fear of losing him on a free transfer looms.
Talks are still said to be ongoing although it remains unclear whether the Dutchman will put pen-to-paper on fresh terms or push for a move to the Premier League at the turn of the year.