Bayern are in the market for a new stopper, and they had been heavily linked with Chelsea's Kepa Arrizabalaga before it was confirmed that he was heading to Real Madrid on loan.
Ajax's Geronimo Rulli was also thought to be close to joining Thomas Tuchel's side before picking up a shoulder problem which has ruled him out for approximately three months.
According to German journalist Florian Plettenberg, Manchester City's Stefan Ortega is currently the main priority for Bayern, with the club bidding to sign a new stopper due to Manuel Neuer's injury issues, coupled with the fact that Yann Sommer has moved to Inter Milan.
However, De Gea is also allegedly being considered, and Plettenberg claims that the stopper has given the green light to move to the Munich giants on a free transfer.
The Spaniard has reportedly held talks with Bayern to discuss a deal, and there will not be a problem when it comes to personal terms for the 32-year-old.
De Gea was also heavily linked with Real Madrid when it was confirmed that Thibaut Courtois had picked up a knee problem which would likely end his campaign.
Meanwhile, a number of Saudi Pro League clubs are said to be keeping a close eye on his situation, with the goalkeeper available on a free transfer following his departure from Manchester United.
De Gea was initially expected to remain at Old Trafford on a reduced salary, but he left when his deal expired at the end of June, with Andre Onana arriving as Man United's new number one.
The former Atletico Madrid youngster won the Golden Glove in the Premier League last season, keeping 17 clean sheets in the competition.
However, the Spaniard faced criticism for a series of high-profile errors, while his struggles in playing out from the back were also highlighted in Erik ten Hag's system.
De Gea is believed to be keen to remain in European football despite the fact that he could likely earn more money by moving to Saudi Arabia's top division.
Sven Ulreich played the full 90 minutes of Bayern's 3-0 loss to RB Leipzig in the German Super Cup on Saturday evening.
Tuchel's side are currently preparing for their first match of the new Bundesliga season against Werder Bremen, with Harry Kane set to feature in Friday's fixture.