The veteran centre-half was thought of as a near certainty to move on this summer.
But the COVID-19 crisis and other circumstances may have led him to change his mind.
"Thank you, I appreciate that. But I am not done yet," he wrote on Twitter to a fan who had thanked him for his goal against Borussia Dortmund in the Europa League four years ago.
While Lovren is no longer first choice, he would still hope to get enough game time to play a productive role as the Reds fight for big trophies such as the Premier League and Champions League.
It is unclear if he only plans to remain for the coming year, or he is close to a renewal with the club.