Leicester boss Craig Shakespeare described the offer as 'well below our valuation', and reports in Italy suggest the Serie A club are preparing to make an improved offer of around 40million euros (£36.1million).
This despite the fact Roma sporting director Monchi said earlier in the week that they had given up hope of luring the player to Rome.
Either way, Shakespeare does not expect Mahrez to become a disruptive influence in the Leicester squad.
"He's been very focused and driven," Shakespeare said. "He hasn't given me one problem where I've had to take him to one side and say, 'Oi, pull your finger out'.
"The commitment and energy levels in terms of training, and the respect that you should give your team-mates, has all been there. He's had a professional attitude."
Asked if he thought Mahrez would still be at the midlands club when the transfer window closes on August 31, Shakespeare replied: "I really don't know. Who knows?"