The Serie A side reportedly tabled a second bid of €35million for the Algeria international, who was named PFA Player of the Year when the Foxes stunned the football world and won the Premier League title in 2015-16.
Leicester are understood to have rebuffed Roma's advances ahead of their season opener against Arsenal on Friday and, speaking at an unveiling news conference for new signing Gregoire Defrel, sporting director Monchi stated it was only prudent for the capital club to direct their attentions elsewhere.
"I'm an optimist in all aspects of life," he said. "Roma is doing everything possible to sign the player.
"If my memory serves me right, the last offer – that was so widely discussed – would have been the highest transfer fee ever paid by this club.
"We're not holding anything back in our attempts to sign the player, but right now we're not expecting a positive response.
"At the same time, we're working on other things. The club is close to the limit right now in terms of the effort we're putting in and I truly believe that the offer we've made is a very significant one.
"Whether we reach an agreement with Leicester, or another team, we will have no regrets."
Meanwhile, Roma have announced a friendly with Chapecoense on Friday, September 1 at the Stadio Olimpico.
All proceeds from the match will be donated to the Brazilian club to assist those affected by the plane crash last November when 71 people – including 19 Chape players – were killed en route to the first leg of the Copa Sudamericana against Atletico Nacional.
On Monday, Chapecoense played Barcelona at Camp Nou in the Joan Gamper Trophy and were captained by Alan Ruschel, one of the survivors from the crash.