The Colombian star has endured a difficult second year at the Spanish giants having found himself on the bench for much of the campaign.
That is far from ideal for Rodriguez, who cost Madrid €85 million, around £67million, following a series of impressive performances at the World Cup in 2014.
He is already looking towards a new club this summer and there would be a host of clubs who are keen on his signature, reports suggest.
They all face stern competition from Europe’s elite sides though, with Juventus and Bayern Munich both also suggested as possible destinations.
A big boost for Arsenal’s hopes would be good relationship they have with Madrid having signed Mesut Ozil from them in a similar big money move back in 2013.
Chelsea though, could be in a strong position too, potentially using Eden Hazard, who Zinedine Zidane is fond of, as a makeweight in a deal.