According to the Daily Express, the 19-year-old midfielder was tempted by reported transfer interest from England and Spain, but believes he is in the right environment in Germany to continue his early development.
Gunners boss Arsene Wenger was believed to be monitoring Draxler's progress in the Bundesliga, and has been weighing up a bid to lure him to the Emirates Stadium.
Chelsea have also been mulling over a £38million offer for the Germany international, as Jose Mourinho is keen to strengthen his Stamford Bridge ranks ahead of the 2013/14 campaign.
But, after signing a new contract with Schalke that ties him to the club until 2018, Draxler admits that he's unlikely to leave during the summer transfer window, though he'll consider a move again in the New Year.
"I've still got to work a lot on myself and improve," he told Schalke's official club website. "I'm staying here because I see the chance of playing frequently.
"I'm going to try to do that as much as possible and then we can reconsider in the new year."
Asked specifically about transfer speculation with rumours of interest from Chelsea and Arsenal - as well as La Liga - Draxler added: "I don't want to name any clubs, but the countries are correct.
"And I don't think any country but these two would make sense for me anyhow."