Having seen his long-term association with Manchester United come to an end earlier in the summer, De Gea is available as a free agent.
While he has been linked with a number of clubs in recent week, the 32-year-old has been in no rush to pen a contract with any admirers as he continues to contemplate his options.
More recently, the Spain international has been identified as a potential addition by a number of clubs competing in the Saudi Pro League, including Al-Nassr.
A goalkeeping mistake contributed to last season's runners-up losing to Steven Gerrard's Al Ettifaq by a 2-1 scoreline in their league opener on Monday evening.
However, it appears that a club that boasts Sadio Mane and Cristiano Ronaldo in their attack will face competition from a European giant over the coming days.
According to Sport, Bayern have instigated negotiations regarding a potential deal for De Gea.
Although much of Bayern's focus has been on acquiring Harry Kane this summer, head coach Thomas Tuchel has been left short between the sticks.
Yann Sommer has made the switch to Inter Milan, while Manuel Neuer is going through the final stages of his rehabilitation from a broken leg sustained at the back end of 2022.
Sven Ulreich deputised for the German Super Cup encounter against RB Leipzig on Saturday evening, only for Bayern to succumb to a 3-0 defeat at the hands of their Bundesliga rivals.
Extra urgency will now seemingly be placed on trying to sign a new number one, yet the report suggests that De Gea's wage demands are currently beyond Bayern.
De Gea had been earning a reported £375,000 per week during the latter years of his time at United and had been ask to significantly reduce that income to remain at the Theatre of Dreams, a request that he rejected.
Bayern had held interest in the likes of David Raya and Kepa Arrizabalaga, but De Gea's Spanish compatriots have instead opted for Arsenal and Real Madrid respectively.