The Italian side are thought to have had at least two offers for the 26-year-old rejected, with some outlets reporting that the number is actually three.
Nevertheless, speaking to the press at the unveiling of striker Gregorie Defrel, Monchi said he is still hopeful of coming to an agreement with Leicester for the winger.
"I'm optimistic in life, by nature. Roma is doing everything possible to take the player," Monchi said.
"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.
"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.
"When we find an agreement, whether with Leicester or another company for another footballer, we will have nothing to rebuke because we are really doing everything.
"The player has expressed his will publicly, but he is right to respect his club. In the end all will be resolved according to Leicester's will to accept it or not."
Mahrez has been linked with a move away from the King Power stadium all summer and recently confirmed his desire to move clubs before the end of the current transfer window.
Nevertheless, Foxes captain Wes Morgan has said that the Algerian's attitude in training has been excellent.
Mahrez announced himself to the world during Leicester's shock Premier League title win in 2015/16 but endured a more frustrating campaign last season, though his stock remains high, hence the interest from the likes of Roma.