The Reds are understood to be targeting the Argentina international as a replacement for Luis Suarez, with an offer of around £18m thought to be on the table.
Lavezzi is rumoured to be reluctant to leave the Parisians after enjoying two successful seasons at the Parc des Princes, but the Ligue 1 champions are determined to get him out.
PSG need to raise money to fund a swoop for Manchester United target Angel di Maria, with the realities of Financial Fair Play setting in after the £50m addition of David Luiz.
Lavezzi is thought to be reluctant to stay if he’s no longer wanted, and Brendan Rodgers is hopeful that he will be tempted by Champions League football on Merseyside.
Italian sides Juventus and Napoli – his former club – have also been linked with him, but they are unlikely to be able to compete with Liverpool financially.
The South American is thought to be seen as an ideal addition by Rodgers, who is attracted to his ability to play on either flank or as a main striker.