The 28-year-old is also wanted by Paris Saint-Germain and Manchester City after making it clear that he does not intend to sign a new contract at the Emirates Stadium, leaving him open to depart as a free agent next summer.
PSG are understood to be leading the way for the Chile international, as they are prepared to offer him in excess of £275,000 a week to join, but Spanish outlet Don Balon suggests that Madrid could make an early move in January.
It has previously been claimed that Los Blancos had been put off making a move due to the player's Barcelona links, having spent three years at Camp Nou before joining Arsenal in 2014.
Zidane is reportedly keen to push ahead with a bid, however, after seeing his side make a slow start to their La Liga title defence.
Madrid have accrued eight points from the first 15 on offer, failing to win any of their three home matches thus far.