Standard Sport understands the 31-year-old, who has a year left to run on his current contract, is being lined up by the Russian champions for a summer transfer.
It is understood that the two clubs are yet to hold talks over a move but Liverpool would be willing to consider a sale for the right price.
Lovren has largely been used in a back-up capacity for the Reds during their title-winning campaign, making 10 Premier League appearances and a further five in all competitions.
The Croatian arrived from Southampton in a £20 million deal from Southampton back in 2014, when Brendan Rodgers was in charge at Anfield.
Lovren has gone on to make 185 appearances for the club, scoring eight goals in that time.
But he faces an uncertain future at Liverpool after falling behind Joel Matip and Joe Gomez in the pecking order to partner Virgil van Dijk in defence.
Zenit, managed by former Russia international Sergej Semak, have clinched the domestic title in Russia and have qualified for next season's Champions League.
The opportunity to play regular football in Europe's premier competition may prove a tempting proposition for Lovren, although the centre-back has suggested he is happy to remain on Merseyside.