Bayern Munich are believed to want more than €10m for the 29-year-old if he leaves this summer.
Nubel spent the past two seasons on loan at VfB Stuttgart and would be expected to become Besiktas's first-choice goalkeeper if he completed the move to Istanbul.
Despite the interest, the Germany World Cup squad member is not currently leaning toward a switch to Turkey and is said to be weighing other options as well.
Reports have previously linked Nubel with Juventus and Manchester City, though at either club he would face significant competition for a starting role.
His salary also presents a complication for any move. Nubel earns up to €11m at Bayern, a figure that Europa League side Besiktas would be unable to match.
The situation leaves Bayern in a difficult position, with the goalkeeper one of several fringe players the club is hoping to move on this summer but facing limited genuine interest at a fee or salary level the club considers acceptable.
Nubel has struggled to establish himself as Bayern's undisputed number one since returning from Schalke in 2020, with Manuel Neuer's continued presence at the club restricting his opportunities and pushing him toward a series of loan moves designed to keep his career progressing elsewhere.
His World Cup involvement with Germany this summer has at least given him a platform that some of his Bayern team-mates currently being shopped around have not had, which may yet help generate fresh interest from clubs assessing goalkeeping options before the window closes.