Compper, a 32-year-old centre back, came on as a substitute in Leipzig’s 2-2 draw with Mainz at the weekend.
And Kicker is crediting Celtic with an interest in the defender, who has found first-team football hard to come by at the Red Bull Arena.
He has played just three times this season, playing the first 45 minutes of a 1-0 defeat away to Augsburg, coming on as a 71st minute substitute in a 2-1 loss to Besiktas in the Champions League as well as playing the second 45 minutes against Mainz.
He is under contract with Leipzig until 2019, but the club may be prepared to let him leave in January for first team football. Leipzig sporting director Ralf Rangnick is reported to be happy for Compper to seek a move away.
Compper began his senior career with Borussia Monchengladbach and has had spells at Hoffenheim and Fiorentina.
He has been capped once for Germany, playing 77 minutes of a 2-1 loss to England in 2008.