After undergoing back surgery at the start of the season, there is a belief Ter Stegen will return to training in mid-December.
However, the Germany international is not in coach Hansi Flick's plans and instead has been encouraged to find himself a new club in January.
Besiktas are making a loan proposal that covers his entire salary until the end of the season, with an €8m option to buy, according to Sport.
Indeed, the Turkish giants, facing difficulties with their veteran goalkeeper Mert Günok, sees Ter Stegen as the ideal solution to stabilise their defence.
Besiktas' sporting director, Eduard Graf, formerly of Borussia Dortmund, is playing a central role in the discussions.
Ter Stegen, whose contract runs until 2028, remains committed to Barça, but a loan appears to be a pragmatic solution to stay in the plans for the German national team ahead of next summer's World Cup.