The Eagles remain keen to extend the 32-year-old's stay at Selhurst Park beyond the summer, with his current deal expiring in June as things stand.
The Senegalese international played a regular role in Patrick Vieira's new-look Palace lineup for much of the 2021-22 campaign, featuring in 27 Premier League appearances in central midfield or at the heart of the defence.
However, with no contract extension being agreed as of yet, Kouyate is free to talk to other clubs ahead of a move this summer, and Turkish outlet Fanatik are reporting that Trabzonspor are one of the sides looking at taking advantage of his current situation.
Karadeniz Fırtınası ended a 32-year title drought to win the Turkish Super Lig eight points clear of nearest challengers Fenerbahce this year, and Abdullah Avci's side are preparing to build on that success in the upcoming transfer window.
The report claims that Trabzonspor were interested in concluding a transfer in the second half of the 2021-22 season, before eventually being put off by a substantial transfer fee that was requested by Palace.
Since signing for the Eagles for around £9m in August 2018, Kouyate has gone on to make 141 appearances for the club across all competitions, scoring three goals.