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The centre-back is one of three stars who is out of contract at the end of the season.
Trent Alexander-Arnold and Mohamed Salah are also yet to sign extensions despite several months of negotiations, although it does look like the forward is on the brink of extending his Anfield stay.
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And although Van Dijk said in March he had 'no idea' if he would remain at Anfield, it now appears that he is likely to remain a Liverpool player next season.
Speaking following Sunday’s 3-2 defeat to Fulham, the Dutchman revealed that there has been 'progress' in negotiations over a new contract.
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He said: “There is progress, yes.” When pressed on whether that meant he would sign an extension, Van Dijk replied: “I don’t know, we’ll see.”
Before he added: “Listen, these are internal discussions, and we’ll see.
I love the club, I love the fans and they were there for us again [at Craven Cottage].
“We wanted to reward them but I ask them to be there again on Sunday [against West Ham] and make the stadium an amazing venue like always for us.”
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And reacting to the development on the It's Called Soccer!
Podcast, Neville said: “That’s great news for Liverpool.
Salah is amazing, Trent is fantastic, but Van Dijk is the most important I think.
“If he signs I think the pressure comes off Liverpool a bit in that they’ve not lost all three.”
Speaking last month, Reds boss Arne Slot made it clear that he wanted his captain to remain at Anfield next season.
For more news relating to West Ham, visit our sister site WHU Latest Live.