Both Turkish giants have spoken to intermediaries about a possible deal for the 22-year-old, according to reports in Turkey, with Galatasaray having had the midfielder suggested to them by agents while Fenerbahçe have gone further and made direct contact with Brighton over his availability.
Neither club has lodged an official offer, and Brighton have made clear he is not for sale, with the Seagulls instead accelerating efforts to tie him down to a new contract.
Ayari's existing deal at Falmer stadium runs out next summer, raising the prospect of Brighton losing him for nothing if fresh terms cannot be agreed. AIK, the Swedish club that sold him to Brighton in January 2023, are shielded from that risk to some extent by a sell-on clause included in the original transfer.
AIK chief financial officer Hakan Strandlund recently confirmed the Stockholm club stand to receive a significant windfall if Ayari is sold on, with previous reporting suggesting AIK are entitled to around 15 per cent of any future transfer fee.
The Swedish international has become an increasingly important presence in Fabian Hürzeler's midfield after loan spells at Coventry City and Blackburn Rovers early in his Brighton career, and he impressed again in pre-season, scoring twice in a 3-0 friendly win over Roma.
Liverpool and Chelsea, who were both linked with Ayari over the summer, are continuing to monitor his situation, though neither club is understood to have made any concrete move.
Brighton's resistance means Turkey remains only a distant possibility for now, with the Super Lig transfer window not closing until Friday, leaving Galatasaray and Fenerbahçe a handful of days to test the Seagulls' resolve before their own market shuts.
For Brighton, keeping their contract security intact is seen as the priority over any short-term sale, whatever the size of the offer that eventually arrives.
Ayari joined AIK's academy in his native Sweden before breaking into the first team and earning his move south, and he has represented Sweden at senior international level having previously come through the country's youth ranks.
His versatility across midfield positions and his composure in possession have made him increasingly central to Hürzeler's plans as Brighton look to build on last season's progress under the German head coach.