The attacking midfielder has long been linked with a move away from the Veltins Arena, with Arsenal and Real Madrid among those said to be interested in his services.
General manager Horst Heldt revealed that Arsenal did submit an official enquiry for the Germany international in the January transfer window, which did not meet Schalke's expectations.
And Heldt believes that the Gelsenkirchen club can keep the 20-year-old beyond the current campaign.
Heldt told Sport1 media: "Arsenal officially enquired about the services of Julian Draxler during the recent transfer window. We did not want him to leave.
"Of course, it was big money on offer, but still it was not enough for us.
"I am convinced Julian will be playing at Schalke past the end of the season. We are able to offer him enough ambition."
Draxler has not featured for Schalke since December because of an ankle injury, but is nearing a return to action after resuming full training.
