National and local Liverpool media have reported that the Reds have now opened talks with the 43-year-old, after he left the Cherries at the conclusion of the season.
Former Bournemouth technical director Richard Hughes is leading the negotiations as sporting director at the Merseyside club.
It was Hughes who appointed Iraola at the Cherries in 2023, when he arrived after leaving Rayo Vallecano in Spain.
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Liverpool are keen to complete the hiring process by June 11 when the World Cup starts, and, as reported, Iraola is the clear front-runner.
He is reportedly keen on taking the job.
Reports have suggested that the Basque boss would be keen on bringing Tommy Elphick with him to Anfield, but the club have not made any approaches for any further additions to the staff.
Liverpool are keen to go back to the front-foot, aggressive football that was played by the club under boss Jürgen Klopp, and they see Iraola as the man to do that.