The Cameroon international will be a free agent in the summer, but reports in Germany suggest the Bundesliga outfit have offered him a new four-year deal with an annual salary of €3.5 million.
However, the 24-year-old has been reluctant to reach an agreement, with Schalke's sporting director Horst Heldt revealing: "It is not going smoothly. On our side everything is clear, but Joel wants to wait and see Schalke's development."
On Thursday, Heldt added: “There are no requests for him. Joel's intention is not to leave the club in January. And if it was like that, we would prevent that with everything we have."
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Schalke coach Andre Breitenreiter is also loathe to lose Matip, as he wants to further the competition for places in his squad.
The 42-year-old insists that he "can exclude that Joel will leave us in the winter,” though if the player continues to stall negotiations, Matip is expected to move to Merseyside.
Liverpool manager Jurgen Klopp is looking for a defender, with 34-year-old Kolo Toure set to leave the club when his contract expires at the end of the season.
The veteran centre-back has been working on his coaching badges and is likely to focus more on the management side of the game, while Klopp knows Matip's strengths and weaknesses well from his time at Borussia Dortmund.