Liverpool are reportedly on the brink of sealing their fifth summer signing.
The 30-year-old is currently on the books at Bundesliga outfit Augsburg.
The report goes on to claim that the Estonia captain - who has 108 caps for his country over a 13-year international career - will pen a two-year deal with the Anfield club this week.
Some Liverpool fans and players might remember him after Jurgen Klopp's men faced the Germans in a Europa League last season, in which the Premier League side won 1-0 on aggregate.
Marko Grujic and Joel Matip both signed for Liverpool earlier this year but completed their moves this summer, while goalkeeper Loris Karius and £30 million man Sadio Mane also joined up with Klopp on Merseyside this summer.
Klopp has also lost two centre-halves in Martin Skrtel and Kolo Toure so the prospective arrival of Klavan is a welcome one indeed.