Arsenal have been strongly linked with a move for the 26-year-old in recent weeks, but it is understood that Roma lead the race, and the Italian giants are confident of striking a deal this summer, according to Monchi.
"I'm optimistic in life, by nature. Rome is doing everything possible to take the player," Monchi told reporters. "I am happy with the effort the club is making to take him. It is close to the limit. I do not think the club has often bid over €30m (£27.1m).
"If my memory does not deceive me, the latest offer would be the club's most expensive purchase in our history. We are, therefore, engaged and expect an answer. But it is true that we are working on other solutions."
Mahrez scored 10 times in all competitions for Leicester last season, with four of those coming in nine Champions League matches as the Foxes reached the quarter-finals of the competition.