Lovren, 31, leaves Liverpool as a Premier League winner after six seasons at the club.
The Croatian international made 185 appearances for Liverpool, scoring eight goals.
According to reports, Zenit are paying in the region of £10.9m (€12m) for Lovren.
Everyone at the Reds would like to thank Dejan for his brilliant contribution over the years and wish him all the very best in his new challenge ❤️
— Liverpool FC (Premier League Champions 🏆) (@LFC) July 27, 2020
Jurgen Klopp praised the outgoing Lovren, speaking of his contributions to the successful Liverpool side.
“Another Liverpool legend who leaves the club, because he was absolutely a very, very important part of this team from the first day since I was in,” Klopp told the official Liverpool site.
“I remember when we played with Dortmund against Liverpool in the pre-season friendly and Dejan scored with a header with incredible power. That was the image I had in my mind when I came here and I thought about the defenders we had at the time.
“He played super games and scored one of the most important goals of our history in the 4-3 against Borussia Dortmund; a really wonderful moment to share.
“He had so many good games. Yes, I think he will not like the Tottenham game when I took him off in the first half, but it is part of football and part of development – and he always came back.
“When he was fit, he was very, very, very often in the team. A great player and a great person.
“Yesterday we said, ‘Who is now the best friend of Mo?’ because they were really real, real bezzies. They were really close and that’s the thing, we will miss him.
“He is a model athlete, has everything you need and now he goes to Russia, but he can come back always – he will be always welcome because he is a really nice guy.”