The Italian outfit had been hopeful of signing the Algeria international at their second attempt, but that ambition has faded in thin air despite improving on their initial bid of £20m.
And despite the 26-year-old disclosing he wants to leave the King Power Stadium, Leicester City manager Craig Shakespeare insisted Mahrez would remain a Fox until the club received a suitable offer.
“Riyad is our player until I am told otherwise from the board that a bid has been accepted,” Shakespeare told media.
“He is turning up for training, has been very professional, got about his business and you see that from his performances.
“Riyad was honest enough to come out at the beginning and say he would like to move to a top six club. He was putting out offers for that.
“We are aware of the situation and until we get an offer that is suitable, he stays with us.”
According to the Serie A sporting director Monchi, Roma will not make a third offer for the winger with Leicester reportedly seeking a fee of £50 million.
Mahrez joined Leicester from French second-tier side Le Havre in 2014 and played a starring role in their Premier League triumph in 2015-16.