Striker Moussa Sylla has emerged as the club's leading sale candidate, sporting director Frank Baumann having made clear that Schalke need at least one significant departure to meet their financial targets.
"Our economic targets as of December 31 haven't been reached yet," Baumann said.
"The summer transfer window runs until September 1, so it's also very likely that we'll need a bigger departure to reach our economic goals."
Sylla scored seven goals and added three assists in 32 appearances last season as Schalke won promotion, and a move to New York City FC worth more than €5m collapsed late in the process back in January; the club would be open to a similar fee this time.
Christian Gomis, signed for around €1.5m only a year ago, has struggled to make an impact in his first season at the club and has been given permission to leave, with a loan to Austria Wien discussed in recent weeks without yet materialising.
Bryan Lasme, praised by head coach Miron Muslic as a "weapon" during pre-season despite a recent fresh injury setback, looks more likely to stay, given Schalke's difficulty finding a taker for his high wages amid his fitness record.
Several other players have effectively been frozen out.
Midfield talent Max Grüger and wing-back Finn Porath have no long-term future at the club but have attracted little concrete interest, while left-back Anton Donkor remains high on the exit list after a proposed move to OFI Crete collapsed when he hesitated over the switch — Schalke would even pay him a severance package to remove his salary from the books.
Martin Wasinski, on loan at RFC Liège last season, is a transfer target for Eredivisie side SC Cambuur, who signed fellow Schalke outcast Ilyes Hamache at the end of June.
Injury-hit defender Tomas Kalas and young full-back Steve Noode round out the list of players with uncertain futures, though no transfer interest has yet emerged for either. Tomas Kalas, Steve Noode, Wasinski, Anton-Leander Donkor and Grüger have all been training with the club's under-23 side rather than the first team as their situations are resolved.