The Algeria international has made no secret of his desire to leave the King Power Stadium during the current transfer window, and Roma are thought to see him as an ideal replacement for Mohamed Salah, who joined Liverpool last month.
The 26-year-old has also been heavily linked with Arsenal throughout the summer, although Arsene Wenger is understood to favour a move for AS Monaco winger Thomas Lemar.
Roma have emerged as frontrunners for his signature as a result, and the Metro reports that the club have now agreed personal terms with Mahrez.
The two clubs must now come to an agreement over a fee, which could prove more complicated with Roma still some way short of Leicester’s £44m asking price.
The Serie A outfit are understood to have offered closer to £26m for the 2015-16 PFA Player of the Year, who scored 10 goals across all competitions for the Foxes last season.