The one-time Bolton loanee has been in excellent form for the Portuguese giants for some time, attracting interest from a number of big clubs across Europe.
Zenit Saint Petersburg were rumoured to have been close to agreeing a deal for the 22-year-old, however their inability to tie up an agreement has alerted the Gunners.
Arsene Wenger is known to be desperate to bolster his side’s attacking options, with a series of injuries having left him short of cover in the final third.
The likes of Alvaro Morata and Dimitar Berbatov appear to be unlikely deals for the London club this month, which has seen Rodrigo emerge as a potential arrival.
Wenger is said to be a fan of the versatile Brazilian-born attacker, who has been told that he can leave Benfica for the right price as they look to cope with financial difficulties.
His price is believed to be around £20m, but Arsenal are exploring the possibility of a loan deal with a view of a permanent transfer once the season ends.
Benfica are believed to be more open to a quick sale, but the prospect of their player impressing in the Premier League could be attractive as his value may rise significantly.
Rodrigo started his senior career at Real Madrid, but enjoyed a brief spell in England with Bolton in 2010/11, making 17 appearances for the Trotters.