The Germany international only joined the Bavarians in a £45m from Manchester City back in the summer of 2020, and has gone on to make 89 appearances for the club across all competitions so far, scoring 24 goals.
However, the arrival of Sadio Mane from Liverpool this summer has reportedly raised concerns in Sane's camp over a potential lack of game time next season, particularly after the winger started just 22 of his 32 Bundesliga appearances during the 2021-22 campaign.
As a result, the 26-year-old's agents have offered him to Real to discover any potential interest in his signature this summer, according to Defensa Central.
The report suggests that Sane's agents raised the question during their visit to Madrid to seal the completion of Luka Jovic's transfer from Los Blancos to Fiorentina.
However, it further outlines how no response has been given as things stand, with any potential deal being hugely dependent on the future of Marco Asensio, who has been heavily linked with a move away from the Bernabeu during the transfer window.
Should Sane remain at the Allianz Arena next season, he will go in search of his third consecutive Bundesliga title with the club, but may have to compete with the likes of Mane, Kingsley Coman, Jamal Musiala, Thomas Muller and Serge Gnabry for a place in Julian Nagelsmann's starting lineup, although Gnabry has also been linked with a move away from Bayern after entering the final 12 months of his contract in recent weeks.