The Germany international is one of his country's brightest talents, and the 19-year-old has enjoyed an impressive season in the Schalke midfield this term.
Reports in Germany had linked Manchester United and Chelsea with a move for Draxler, while Serie 'A' giants Inter Milan and AC Milan were also thought to be interested.
But Schalke general manager Horst Heldt is hopeful that qualification for next year's Champions League will persuade Draxler to stay put.
Speaking to German newspaper Bild, Heldt said: "Julian Draxler is under contract at our club and he is an important player.
"We want to keep him at our club for as long as possible. It's our obligation to ensure he feels fine being with us.
"From our point of view, it would be helpful to win a berth in the UEFA Champions League for next season. This would be a boost!"
Draxler is under contract with the Bundesliga club for another three seasons. Schalke currently sit fourth in the German top flight, occupying the final Champions League qualifying spot.
They were knocked out at the last-16 stage of this year's competition, as a Burak Yilmaz inspired Galatasaray progressed from their tie into the quarter-final stage.
Schalke lost Lewis Holtby in the January transfer window to Tottenham, but that loss was partly softened by the rise of Draxler in the first-team.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms