Despite competition from the Premier League, PSG have stepped forward and already struck terms over a contract with the Georgia star's agent.
Le Parisien says things are moving quickly and PSG are confident reaching a deal with Napoli within the next 48 hours.
A fee of around €75 million (£63 million) has been mentioned for Kvaratskhelia.
PSG could include a player in their offer to accelerate negotiations and reach an agreement before English rivals have the chance to get involved.