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The Gunners are said to be closing in on the addition of Denis Suarez from Barcelona, and it had been suggested that the deal would represent the Premier League's side biggest during the current transfer window.
However, the Catalan giants could now block that move and according to Don Balon, Arsenal have made an approach for Colombia international Rodriguez.
The 27-year-old is currently in the second of a two-year loan spell at Bayern Munich, but it has been claimed that the German outfit would not prevent an exit before the end of this month.
The report also indicates that Real have already accepted a bid of £63m from Arsenal, leaving open the possibility of a deal being finalised in the coming weeks.
Rodriguez has made just 11 outings in Bundesliga and Champions League fixtures this season, with his last appearance coming at the start of November.
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