The Colombian international has been told he can leave the Bernabeu this summer and Real Madrid are willing to listen to offers of around £65m.
The Gunners are hoping family ties will give them the edge with Rodriguez’s brother-in-law, David Ospina, 27, currently at the Emirates.
Rodriguez has fallen from favour in the Spanish capital since Zinedine Zidane took over as coach and is open to a move to the Premier League.
His agent Jorge Mendes has made both London clubs aware of his availability but they face competition from Paris St Germain.
Whilst Arsenal’s most pressing need is for a central defender, Arsène Wenger is well aware that his forward line needs improving.
Olivier Giroud, 30, proved inconsistent last season and failed to score for 15 games, scuppering the Gunners title bid just as they had hauled themselves to within two points of eventual champions Leicester City.
Danny Welbeck, 24, is out until April 2017 with injury and Alexis Sanchez proved ineffective in the role during Sunday’s defeat to Liverpool.
Having missed out Gonzalo Higuain, 28, and Alvaro Morata, 23, this summer, Wenger is struggling to make signings to strengthen his squad. Talks with Lyon over Alexandre Lacazette, 25, have stalled and winger Riyad Mahrez, also 25, has signed a new four-year deal with Leicester.
A move to north London may suit both parties as Rodriguez looks to inject some life back into his career.
Read all transfer ins and outs, completed deals visit football transfer news.