According to Manchester Evening Newsreport that Pep Guardiola has told Manchester City chief executive Ferran Soriano and sporting director Txiki Begiristain to wrap up the deal for Mahrez in time for the club's USA pre-season tour.
Jorginho made 35 appearances for Napoli this season, scoring three times.
The newspaper reports that an initial €45m offer was rejected by the Partenopei, who are holding out for €55m.
And sources in Italy claim that the clubs are moving ever closer to a basic £42m agreement with the possibility of a further £1.75m going to the Italians in add-ons.
The 26-year-old midfielder has been identified as one of Manchester City's priority targets, but the club have found Napoli to be tough negotiators in the market. If Napoli and City find a solution, we can talk about it.