The North London club had been linked with a move for the Germany international this summer, though Borussia Dortmund have managed to keep hold of their man, who picked up an injury ahead of the World Cup.
However, the 25 year old will be allowed to leave for £20m at the end of the campaign, with the Gunners ready to pounce.
Arsene Wenger added Danny Welbeck to his squad of transfer deadline day, having already brought in Calum Chambers, Alexis Sanchez, Mathieu Debuchy and David Ospina in the close season.
They have begun the campaign by taking five points from their first three league matches, drawing with Everton and Leicester City having beaten Crystal Palace in their season opener.
Their next match is against champions Manchester City at the Emirates Stadium a week on Saturday, when they can be backed at 15/8 to take all three points.