Athletic hope to take the midfielder on-loan and will seek to close the deal once their Champions League status is decided.
If they overcome Napoli in the second-leg of their qualifier, Athletic will take Illarramendi off Real's hands for the season.
Real management know that at 24, Illarramendi needs to be playing regularly and sees 12 months away on-loan as the best option.
The move to Athletic, however, will create some problems for Real president Florentino Perez, who is still trying to mend the bridges he burned with Real Sociedad last summer upon ferrying Illarramendi away from San Sebastian.
For the big stories and the banter follow us on Twitter: @tribalfootball