The Gunners have been linked with a move for Rodriguez in the past and are said to have been keen on landing the 19-year-old in the summer as he goes in search of the first team football that Real Madrid cannot guarantee him.
While Madrid are reluctant to sell, the talented youngster has agreed a deal to join Benfica on a long-term contract according to Record, which could be finalised by the end of the week.
Rodriguez is out of contract in the summer, and would command no transfer fee should he complete a move to either Arsenal or Benfica, although Madrid would be entitled to compensation.
The Gunners were said to be particularly keen on landing the former Spanish U-19 international, who made his debut for Los Blancos last month, in part because of his availability without a transfer fee.
Rodriguez has garnered comparisons to Cristiano Ronaldo in the past thanks to his tendency to attack at speed from wide positions, and has earned the nickname of ‘Cristiano of Valdebebas' after the name of Madrid’s training ground.