The Gunners have lacked a strong central presence since Santi Cazorla suffered an ankle injury in October 2016. The Spaniard has suffered repeated complications since then, and it is not yet known when he will return, or in what condition.
In light of that, Arsene Wenger has made signing a midfielder his number one priority and Modric is believed to be his primary target, despite the Croatian's history with Arsenal's arch-rivals Tottenham.
Modric made 159 appearances for Spurs in four years at White Hart Lane, scoring 17 goals. He was the club's player of the season in 2010/11 and is fondly remembered by the Tottenham faithful.
A report from Don Balon claims that Modric has informed Real Madrid president Florentino Perez of his desire to leave the Bernabeu as he seeks a new challenge in the twilight years of his career. AC Milan are also rumoured to be interested in his signature.
However, Madrid's lack of a suitable replacement means that they are unlikely to allow Modric to leave before his contract expires in 2020.
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Modric has been a regular for Real Madrid this season, but has missed the club's last five games with a muscular injury. He may return for this week's crucial Champions League second leg against PSG, with Real leading 3-1 on aggregate.
If he were to join Arsenal, Modric would become the first player to cross the north London divide since Emmanuel Adebayor, who left Arsenal in 2009 and joined Spurs in 2011.
More famously, Sol Campbell moved from Spurs to Arsenal in 2001, and was subsequently branded 'Judas' by the Tottenham fans.