The Gunners are in the market for midfield reinforcements this summer, with Real Madrid loanee Dani Ceballos increasingly likely to return to his parent club at the end of the campaign.
Arsenal manager Mikel Arteta is understood to have earmarked Rodriguez as his top target to replace Ceballos, and Estadio Deportivo reports that the two clubs are now close to reaching an agreement.
The 27-year-old has a release clause of £69.5m in his contract at Betis, but the Spanish outfit's financial difficulties mean that they are likely to be forced to sell him at a reduced price.
Arsenal could still splash around €50m (£43.3m) on the Argentine midfielder, though, earning Betis a significant profit having only spent around £6m on him last year.
Rodriguez is a defensive midfielder and has featured 31 times for Betis in La Liga this season, scoring one goal and amassing 10 yellow cards.