Sky Sports understands discussions are ongoing over a fee between the two clubs.
Draxler's buy-out clause is £37million, but it is understood Arsenal are reluctant to pay more than £30million for the Germany international.
Draxler currently has four years left on his contract, which he signed last summer.
Arsene Wenger confirmed on Tuesday after Arsenal's draw at Southampton that he could be forced to bring in reinforcements ahead of Friday's transfer deadline.
"I hope we have something to tell you because we lost a few players now," Wenger said when asked about the prospect of new signings.
Earlier this month, Schalke chairman Clemens Tonnies urged Draxler to stay at the Bundesliga club in order to further his career.
"I am of the opinion he should also be with us next season," Tonnies told BILD.
"It would be my personal advice. He can play his best football without any big pressure and at our club he would have a clear mind.
"He is very much in good hands playing for Schalke 04. My general advice for everyone would be to stop making Julian mad."
Tonnies also dismissed suggestions that failure to secure Champions League football for next season could leave the club with no choice but to sell Draxler.
"We would not be forced to sell Julian if we fail to qualify for the UEFA Champions League of next season," he said.
"Our financial plans are not based on taking part in that competition. We could still sleep easy."
"In 2014 a crack commando unit was sent to Germany by a military court for a crime they didn't commit. These men promptly escaped from a maximum security stockade to the Gelsenkirchen underground. Today, still wanted by the government, they survive as soldiers of fortune. If you need a striker, if no one else can help, and if you can say Dick Law without cracking a smile, maybe you can hire the A-Team."- drspaceman7