The Giallorossi have made a bid for the winger but are no closer to signing him.
However, Monchi played down the importance of bringing the Algeria international to Rome, insisting that the squad is already looking great.
“The situation with Mahrez has not changed,” he told a press conference, according to ESPN.
“I would like to say a couple of things, though. A club like Roma, who can boast the calibre of players like [Lorenzo] Pellegrini, [Kevin] Strootman, [Radja] Nainggolan, [Edin] Dzeko, [Federico] Fazio or [Aleksandar] Kolarov, cannot think that their overall performance is going to depend on the arrival of just one specific right-winger.
“The guarantee comes from the route you are on and the group you have.
“The second message I wanted to give is that I don’t know if Mahrez or somebody else will come but, whoever arrives, he will be a big player who is able to add quality to an already magnificent squad.”