With the deal for Real Madrid's Sami Khedira stalling because of his wage demands, Schneiderlin is seen as a more affordable target.
Gunners' manager Arsene Wenger has recently strengthened his attack, with the £35m capture of Alexis Sanchez from Barcelona, and his defence, with the £12m purchase of Mathieu Debuchy from Newcastle United, but buying a defensive midfielder still features heavily in his summer plans.
Southampton have had their squad ripped apart since the end of last season. Manager Maurico Pocthettino has joined Tottenham, Adam Lallana and Rickie Lambert have moved to Liverpool, and Luke Shaw has gone to Manchester United.
As well as the potential departure of Schneiderlin, defender Dejan Lovren is reportedly closing in on a £20m switch to join Lallana and Lambert at Anfield.
Schneiderlin spoke of his desire to leave Southampton last month.
"I've been there for a long time now," the 24-year-old France international had said. "We'll see how things go, but the directors know my position and what my intentions are.
"Everyone has ambitions, everyone has his own future in his hands, so we’ll see. Every player wants to play on big occasions."
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