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The Colombian appears destined to leave Madrid this summer, having fallen so far down the pecking order under Zinedine Zidane that he was left out of the squad entirely for the Champions League final.
He had been heavily linked with a move to the Premier League since the turn of the year, with Manchester United and Jose Mourinho the most likely suitors.
But now Spanish television show El Chiringuito have revealed that it is Arsenal who lead the race for his signature, having tabled an offer that falls just short of their €80 million (£69.8m).
The only other club to table an official bid are Italian giants Inter Milan, though their offer is also too low for Madrid to consider accepting at this time as they look to incite a bidding war.
Rodriguez already has ties to north London, having married Arsenal goalkeeper – and Colombia team-mate – David Ospina’s sister.
Arsene Wenger is reportedly ready to spend big this summer as he eyes a genuine title challenge following his decision to remain in charge at the Emirates for another two years.
He has already been linked with a record-breaking bid for Monaco’s Kylian Mbappe, and is also lining up deals for Lyon striker Alexandre Lacazette and Nice midfielder Jean Seri.
It would not be the first time Wenger has raided Real for one of their cast-offs, having signed Mesut Ozil from the Spanish giants four years ago.
Rodriguez, meanwhile, is desperate for regular pitch time, especially ahead of next summer’s World Cup where he looks to defend his Golden Boot, and has already warned that he will leave Real to get it.
‘I can’t assure I will stay at Real Madrid,’ he said after Real won the Club World Cup in December. ‘I have offers and seven days left to think about my future.
‘I am happy in Madrid, but I want to play more. I have a bitter feeling as I wanted to play the final, but I am happy for the title. It’s the 15th title in my career.’