The Senegal international struggled to live up to expectations after completing a move from Liverpool last summer.
Mane's 2022-23 campaign was disrupted by injury and a dressing room bust-up with teammate Leroy Sane following Bayern's Champions League meeting with Manchester City in April.
Despite having two years left to run on his contract, Mane's time with the German club looks set to come to an end after just one season.
According to journalist James Benge, Al-Nassr have submitted a €40m (£34m) bid and are now close to completing a deal.
The Saudi Pro League side have submitted a request to the Player Acquisition Centre of Excellence at the Ministry of Sport for the financing of the potential transfer.
The report claims that the former Liverpool man has been offered a lucrative contract that would see him earn in excess of £650,000 a week.
Should the deal go through, Mane will link up with Cristiano Ronaldo, David Ospina, Marcelo Brozovic and Seko Fofana.
Al-Nassr also recently completed the signing of Manchester United left-back Alex Telles for a fee believed to be €10m (£8.7m).
Despite enduring a tumultuous season with Bayern, Mane still managed to win the Bundesliga title and the German Super Cup.
Mane, who has earned 97 caps for Senegal, made a total of 38 appearances for Bayern last term, scoring 12 goals and providing six assists.
Prior to joining the German club, Mane spent six years at Liverpool, where he scored 120 goals and 48 assists in 269 appearances.
During his time with the Reds, won the Premier League, Champions League, FA Cup, EFL Cup, UEFA Super Cup and FIFA Club World Cup.