Meyer is currently away with the German national team at the Under-21 European Championship in Poland, where he scored in his side's opening win over the Czech Republic on Sunday (18 June).
But after a season where his performances were criticised by now former Schalke head coach Markus Weinzierl, his future at the club has come under severe scrutiny.
The 21-year-old has just one more year remaining on his current contract at the Veltins-Arena and told Bild before linking up with Germany that he and his management team had decided to reject the offer of a new deal.
There could yet be hope for a positive resolution, however. Following Schalke's recent appointment of Domenico Tedesco to replace the outgoing Weinzierl, Meyer says he will sit down with the new manager before deciding his next move.
"Sure, there is hope that things will be better for me at the club," he was quoted as saying by ESPN. "I have not said I will leave Schalke, but rather that I won't extend my contract. We will see about the rest."
Meyer has already said he will not seek a move to Bundesliga rivals Wolfsburg, but interest from the Premier League reportedly remains following Tottenham's failed efforts to sign him last summer.
Speaking last November, Meyer revealed Schalke had been approached by the Premier League side, saying: "I know there was interest and the club was asked, but Schalke immediately put a stop to it so the matter was settled. Of course I have dealt with it. I know what I have at Schalke and I like to play for this club."
According to Bild, that offer from Tottenham during the final days of the summer transfer window was worth €45m (£39m, $50.6m).
After a disappointing 2016-17 campaign and with one less year to run on his contract, Meyer's value has dropped since then. As reported by the German publication in April, Schalke are now more receptive to the idea of cashing in on the player, suggesting a fee of €20m may now be enough to tempt them into doing a deal.
But with Meyer now seemingly open to seeing out the remainder of his contract at the club, Tottenham and any other interested parties may have to wait.