A delegation including the Saudi Arabia club president Waleed Muaath has been in London this week discussing the possibility of deals with several players including Mahrez.
The Algeria forward has two years remaining on his existing agreement with City but sources have suggested he has become frustrated with a lack of playing time under Pep Guardiola this season.
Mahrez made 22 Premier League starts in the 2022-23 campaign but has been on the bench for much of his side's Champions League knockout matches. His last European start came against RB Leipzig on Feb. 22.
Al Ahli are therefore hopeful of pulling off a transfer coup and convincing Mahrez to move to the Middle East. Karim Benzema signed for Al Ittihad on Monday, joining Cristiano Ronaldo in the Saudi Pro League while N'Golo Kante is set to become another high-profile addition to the league.
The source involved said Al Ahli have indicated they would offer Mahrez €50m+ bonuses per season on a two-year contract with the option of a third year.
Mahrez is thought to be open to moving to Saudi Arabia but it is unclear at this stage whether City would allow the player to leave and if so, the fee they would command. The situation is likely to be addressed after Saturday's Champions League final against Inter Milan.