The deal to take him to Bayern included an £11.5m loan fee with an option to make the transfer permanent for £39.5m at the end of the current season.
However, the initial fee is being paid in installments and the report claims Bayern are willing to let Arsenal take him off their hands if they pay the final £3m which is due to Real.
The Gunners are struggling to raise funds for permanent transfers in the winter window so it could represent an ideal solution for both clubs.
After an excellent start to life under Emery, they have fallen off the pace in the race to qualify for next season’s Champions League and have just one win in their last four Premier League games.
James Rodriguez has been linked with Arsenal and could be available at a knockdown price (Image: GETTY)
Rodriguez is out of favour at Bayern Munich and keen to leave in search of first-team football (Image: GETTY)
The Colombian won the Golden Boot at the 2014 World Cup (Image: GETTY)
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In their 1-0 defeat at West Ham on Saturday, Emery’s side looked devoid of creativity and playmaker Mesut Ozil was left out of the matchday squad, with the boss citing tactical reasons for his absence.
Emery has been publicly critical of the German all season and lacks competition in his position.
While Rodriguez has struggled since moving to the Bundesliga, he would arrive at the Emirates with vast experience at the very highest level.
Unai Emery's Arsenal are struggling and the manager is keen to being in some new faces (Image: GETTY)
Arsenal: Unai Emery on January transfers
He won the Golden Boot at the 2014 World Cup where he inspired Colombia to the quarter-finals and was named in the team of the tournament.
Rodriguez was also named in the La Liga team of the year in his first season at Real Madrid after moving from Monaco in a £40m deal immediately after the World Cup.
And despite starting only five games in the Bundesliga this season, the attacking midfielder has bagged three goals in those appearances.