Acuna is on the books of Real Madrid, but has spent the past season on loan at Girona and has impressed in the Spanish second division.
Girona have an option to sign the 24-year-old on a permanent basis, but missed out on promotion to La Liga after losing to Almeria in a play-off.
Meanwhile, Premier League newcomers Crystal Palace, along with Championship sides Middlesbrough and Nottingham Forest, are also reported to be keen on Acuna.
However, it appears that Acuna is likely to get his chance to play in Spain's top flight after Espanyol stepped up their interest in the forward.
Espanyol scouts have filed several positive reports on Acuna throughout the season and a deal could be finalised this week.