Mahrez has already announced his decision to leave the Foxes this summer, eager to pursue to a fresh challenge a year on from the club's historic Premier League title success.
Arsene Wenger revealed in June he was considering a bid for the Algeria international, while Roma have already made an offer for the player, reported to be worth €23m (£20.6m, $26.8m), that was "politely declined" by the Premier League side.
Reports in Italy have suggested Roma have already agreed a deal with the player and are confident an improved offer of €30m will bring the 26-year-old to the Stadio Olimpico. That offer could fall short of Leicester's valuation of the player, however, with Leicester asking for an offer between €35m and €40m.
But with no official word on a deal, Leicester have now given both Roma and Arsenal an ultimatum, according to Il Messaggero: Either meet their demands before their Premier League campaign officially kicks off on the 11 August or forget about the transaction.
Coincidentally, Leicester begin their Premier League campaign on that date away to the Gunners.
Roma, who completed the signing of Manchester City defender Aleksandar Kolarov on Saturday, are keen to bring in a replacement for Mohamed Salah, who completed a club record move to Liverpool earlier this summer.
Arsenal meanwhile have made Monaco star Thomas Lemar their next priority signing, having already brought Alexandre Lacazette to the club for a club record fee. Monaco, however, have already rejected two bids from the Gunners for the 21-year-old and are determined not to let another one of their prized assets leave following the departures of Bernardo Silva and Tiemoue Bakayoko, who have joined Manchester City and Chelsea respectively.
Mahrez is believed to be Arsenal's next option down the line. "Have we made a bid for him? No. Not yet. Not yet means it could happen, it could not happen," Wenger commented on the Leicester star in June. "It's been a different season this year but it doesn't take anything away from his qualities."
Arsenal have made just two additions to their squad to far this summer in Lacazette and Sead Kolasinac.