Ayari, 22, joined Brighton from AIK in January 2023, in a deal that included a sell-on clause for the Swedish club should he be transferred in the future.
The Sweden international scored three goals and provided three assists in 29 Premier League appearances last season, and has continued that form into the new campaign with a goal in three league starts so far.
His current contract with Brighton runs until next summer and the club have been working on fresh terms to keep hold of him, with Liverpool and Chelsea both tracking his situation at the time.
The Athletic reports that discussions between the two parties are ongoing without any guarantee he will commit his future to the club, with sources suggesting it remains unclear whether he will sign an extension before the season progresses further into the autumn.
Turkish clubs Galatasaray and Fenerbahce have also been credited with interest in Ayari in recent weeks, though Brighton are said to have rebuffed any approaches from Turkey as they focus on tying him down to a new long-term deal.
The newspaper also reports that Brighton hold an option to extend Ayari's contract by a further year, which the Premier League club could activate should they fail to reach an agreement on fresh terms before his deal runs down next summer.
Ayari, who also had loan spells at Coventry City and Blackburn Rovers earlier in his career before establishing himself at the Amex, has scored five goals and provided two assists in 25 senior caps for Sweden to date, becoming a regular under the current national coaching staff.