Jorginho has impressed this season for the Partenopei - who pushed Juventus all the way in the race for the Serie A title - with four goals and four assists in 37 appearances.
Mahrez, meanwhile, has long been pining for a move away from the King Power Stadium and was coveted by the Citizens in January, only for a prospective move to break down at the end of the transfer window.
According to the Daily Mail, City remain favourites to sign Jorginho, valued at £53m by his parent club and also thought to be on the radar of Liverpool and Manchester United.
The report claims that the Citizens have also revived discussions over Mahrez, thought to have a £60m price tag, though the Foxes want winger Patrick Roberts as part of the transfer.
City have broken numerous Premier League records en route to the title and are one win away from an unprecedented 100-point tally.