The promising playmaker completed his long awaited switch to Old Trafford yesterday, after the Red Devils had tracked him for well over a year.
Bilbao were far from keen to be without their man, but eventually lost him as the player bought himself out of his contract with the Basque side.
With almost £30m in their coffers following the exit of Herrera, the La Liga giants are desperate to get some business done, and have set their sights on Arteta.
The 32-year-old is now in the final 12 months of his contract at the Emirates Stadium and looks unlikely to be handed a new deal by Arsene Wenger.
Bilbao are confident that a £3m offer will be enough to lure the playmaker back to his homeland.
Arteta previously played for the San Mames outfit’s regional rivals Real Sociedad, yet that is thought to be a concern the club are ready to work around due to his talents and experience.