The Italian outfit were reported to have placed an £32m offer - representing a record fee for the club - for the Algeria international earlier in the week, which proved unsuccessful.
Monchi has said that Roma will not be tempted into testing the Foxes' resolve further, as "the ball is in [Leicester's] court".
"In the last press conference I said we had bid what we believe to be a fair offer, which is also the biggest in the club's history. We won't bid again," he told a press conference ahead of a friendly with Celta Vigo this weekend.
"We've conducted ourselves in the right way by talking to Leicester City. Now the ball is in their court.
"We're waiting for an answer. We're working on other options but we're relaxed given our squad is 90% complete."
Foxes manager Craig Shakespeare said on Wednesday that he plans to use Mahrez in the Premier League season opener against Arsenal this evening, describing him as "driven" and "focused" in the midst of his transfer saga.