Sky Sports in Italy understand a third offer of £31.8m - a Roma club record - has recently been rejected by the Foxes, following previous bids of £20m and £27m.
"I'm optimistic in life, by nature. Roma is doing everything possible to take the player," Monchi said at the unveiling of another new signing, Gregoire Defrel.
"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 Euros.
"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."