Iraola made more than 500 appearances for the club in the Basque country, which has a rich history and tradition of only selecting players from the region.
With the Cherries boss yet to sign a new deal in Dorset, and his contract running out in the summer, he is reportedly one of the frontrunners to take over at San Mames, according to TalkSport.
He will reportedly inform Bournemouth of his decision by the end of the international break.
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His former assistant at Rayo Vallecano, Inigo Perez, who also represented the club as a player, is another being considered, according to MARCA.
Perez is currently boss at Vallecano, where he took over not long after Iraola departed for Bournemouth, with the intention initially for him to come as his assistant, but work permit issues denied him that opportunity.
Other managers reportedly on the list to replace Valverde include Celta Vigo coach Claudio Giraldez and current USA head coach Mauricio Pochettino.