Sacked Huddersfield Town boss David Wagner is a candidate for the job at Schalke.
BILD says Wagner is in the frame as interim coaches Huub Stevens and Mike Büskens will move on.
Departing Borussia Monchengladbach coach Dieter Hecking is favourite for the job, though Wagner is also being mentioned.
Wagner said earlier this month: "The German Bundesliga is very interesting for me, because I was born and educated in Germany.
"I have never worked in the Bundesliga. That alone would make it very exciting.
"I have two markets for which I feel absolutely ready."