Klavan made more than 100 appearances for Augsburg after joining them in 2012 and his experience is seen as ideal to replace Kolo Toure.
Arjen Robben will be offered a new Bayern Munich contract if he enjoys a successful season, the club's CEO Karl-Heinz Rummenigge has said.
Dortmund defender Neven Subotic's move to Middlesbrough appears to be in doubt following his medical, although his agent has played down the reports.
Bayern Munich boss Carlo Ancelotti says Robert Lewandowski will not be allowed to join Real Madrid this summer.
Premier League champions Leicester City began their preseason campaign with a 2-1 win over Oxford United on Tuesday.
Carlo Ancelotti is excited by the new Premier League season, and has also backed Mario Gotze at Bayern.