Having steered Fulham away from the Championship's drop zone in his first half-season in charge, manager Slavisa Jokanovic will be keen to put his own stamp on the Cottagers' squad.
A move for Blackburn Rovers winger Ben Marshall was mooted by the Sun on Tuesday, and the Star claims Klavan is also on Jokanovic's shortlist.
The 30-year-old Estonia international, who has 108 caps to his name, has a year to run on his Augsburg contract.
Everton reportedly made contact over Klavan last summer, with the player's agent later revealing his client's dream to play in the English Premier League.
But Klavan, who played against Liverpool in this season's Europa League, may have to lower his sights a little - at least for another year.
Fulham are set to lose Dan Burn on a free transfer this summer and fellow centre-back Shaun Hutchinson could follow, though his contract includes the option to extend it by a further 12 months.