Uzun is nominally pencilled in as a fixture in Frankfurt's attack for the coming season, but club officials have not publicly declared him unsellable.
Galatasaray have opened talks with the forward's representatives in recent weeks, with Frankfurt currently demanding at least €60m for the 20-year-old, a figure well above the roughly €40m the Turkish champions are said to be able to offer.
Eintracht Frankfurt's hierarchy would much prefer to keep hold of their prized talent for a while longer, though, and Kicker reports there are "at least internal considerations, contrary to past practice, to put up a stop sign" over any Uzun departure.
Such a stance would only be realistic in the event of one or two other significant sales, with the magazine naming Jean-Mattéo Bahoya, 21, and Farès Chaïbi, 23, as players who could together fetch a fee "beyond €50m."
Sporting director Markus Krösche addressed the speculation directly a few days ago: "The number ten position isn't even a topic right now.
"There are always some reports coming out of Turkey where I sometimes think, is he still here or has he already left?
"At the moment there are no signs that he's leaving us. The way Adi Hütter wants us to play, we can put him in the positions where he can cause the most danger."
Uzun joined Frankfurt from Nuremberg in 2024 for around €11m and has scored 15 goals with nine assists in 59 competitive appearances since, on a contract that runs until 2029.
Beyond Galatasaray, several Premier League clubs and Italian sides including AC Milan, AS Roma and Napoli have also been credited with tracking his progress, though Frankfurt's asking price is understood to be a deterrent for now.
Uzun made his breakthrough as a Turkey international last season and remains regarded as one of the Bundesliga's most exciting young creative talents heading into the new campaign.