The creative midfielder only joined Los Blancos in 2013 from Real Sociedad, but despite his £24m price tag he has failed to hold down a place in the club’s first XI.
Dubbed the ‘new Xabi Alonso’ earlier in his career, the 24-year-old has been lauded in the past for his range of passing and composure on the ball.
And reports now suggest that Arsenal boss Arsene Wenger is ready to offer the playmaker a fresh start at the end of the season after registering a bid for his services.
Illarramendi is thought to be happy with the proposal, and may even accept the transfer this month instead of waiting until the summer.
Life in London is thought to appeal to him, according to the report.