The Giallorossi had identified the Algeria international as one of their primary transfer targets this summer, but they have had three bids rejected for the 26-year-old, the latest of which was a club-record £31.8million offer.
Sporting director Monchi, who only joined Roma from Sevilla earlier this year, has now confirmed the club will not be making any further bids for Mahrez, with the ball now firmly 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," Monchi said. "We won't bid again. We've conducted ourselves in the right way by talking to Leicester City. Now the ball is in their court."
Monchi insists Roma already have a good squad at their disposal and they have alternative options if Mahrez ultimately does not move to the Stadio Olimpico.
"We're waiting for an answer," he added. "We're working on other options but we're relaxed given our squad is 90% complete."