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The Reds have an option to tie the Croatian down until 2022 as part of a clause in his contract and are now set to do so, as per BBC Sport.
However, despite planning to trigger the extension, it appears as though the club are still keen to offload the centre-back but they are able to command a higher fee if they take up the extension option.
That's the word from Neil Jones from Goal, who also reports that both Zenit and Roma are both potential suitors after showing interest.
Lovren joined Liverpool from former feeder club Southampton in a £20 million deal back in July 2014 and has gone on to make 181 appearances for the Anfield club.
The former Lyon man was part of the side that won the Champions League last year, where he will also be receiving a Premier League winners medal after making 10 league appearances for Jurgen Klopp's title-winners this term.
He has, however, fallen behind Virgil van Dijk, Joe Gomez and Joel Matip in the pecking order at the heart of backline.
"It has been a test for me," Lovren told Sky Sports last month.
"But in these moments, I think you should stick with the team even more, to show your quality even if you are not on the pitch with the first XI. The next day in training, you should give everything, just to respect your team-mates.
"Of course, it's not the happiest moment when I see myself on the bench, but it's the decision of the manager. I respect it and I am always the guy who will work 200 per cent behind closed doors."
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