Mahrez has been offered a deal worth around £865,000-a-week and is keen on the move.
It is understood the Saudi Pro League club are willing to pay £30m for the 32-year-old.
Mahrez has two years remaining on his contract at the Etihad, which is worth more than £6m-a-year.
The Algeria star scored ten goals in 41 appearances for City in their treble-winning 2022/23 campaign.
It comes after Liverpool captain Jordan Henderson was offered £700,000-a-week to join Steven Gerrard's Al Ettifaq in Saudi Arabia.
The England midfielder has two years remaining on his current Liverpool deal, which is worth £200,000-a-week.
After 12 years with the Reds, Henderson remains a key player and a vital leader in the dressing room.
It is understood he will hold talks with manager Jurgen Klopp over his future in the coming days.
Liverpool midfielder Fabinho is also the subject of interest from Saudi Arabia.
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Al Ittihad have been one of the biggest movers so far, bringing in French pair Karim Benzema and N'Golo Kante on free transfers, while splashing out £25m to sign winger Jota from Celtic.
And they have now identified their next possible recruit, with The Athletic reporting they will make a £40m proposal to Liverpool for Fabinho.