Sky Sports claim Pep Guardiola has made the offer as he looks to bring the Algerian to the Etihad.
Mahrez handed in a transfer request at Leicester yesterday as City had a £55m bid rejected.
The player City have offered the Foxes is not known but the deal is said to be worth around £65m.
Mahrez has been in fine form this season, netting nine goals and providing nine assists in all competitions.
The 26-year-old is frustrated he hasn't been allowed to leave like former team-mates N'Golo Kante and Danny Drinkwater.
How Manchester City could line-up with Riyad Mahrez
GETTY
“We don’t have left-footed players in that position”
Man City boss Pep Guardiola
Guardiola is looking to Mahrez following an ankle ligament injury to Leroy Sane.
The winger is expected to be out of action for around six weeks.
Guardiola dropped a hint on potentially signing Mahrez earlier today.
He said: “We don’t have left-footed players in that position or guys to push up like full-backs but we are going to adapt.
GETTY
“Right now, we have just three strikers plus Brahim Diaz. We don’t have more for the next month or month-and-a-half because of Leroy and Gabriel Jesus, who is coming maybe in the next five or six weeks.
“We are going to overcome the situation. We are going to adapt the way we play. If David Silva or Kevin De Bruyne have to play as a winger, they will play as a winger. We can adapt several positions to do that.
“Raheem Sterling can play at left-back. We have Raz and Bernardo Silva can play like a winger. In that moment, my first idea was that these guys are going to play.
“For a lot of games, I haven’t known what is going to happen. David is here sometimes; sometimes, he is in Valencia. We are going to decide day by day what is best.
GETTY
“It’s not an ideal situation but it’s how you handle bad moments.
“The important thing is the team spirit and that people are still fit. We have to be focused on what we have to do, game by game.
“We don’t have too many options to play every three days. We have a lot (to do) but we are going to find a solution.
“We play West Brom (on Wednesday), then we have two days of recovery and we are going to one of the toughest places in the Premier League in Burnley. We will try to move forward."