The 52-year-old is leaving the Anfield club in the summer and may head for the Saudi Pro League.
Gerrard's side, Al-Ettifaq, are looking to beef up their coaching staff ahead of next season.
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Per The Mirror, Achterberg leaves with well wishes after serving Liverpool for the past 15 years.
The former Tranmere keeper has worked at the club since 2009, when he joined under Rafa Benitez.
He is a popular member of staff at Anfield, helping to bring Alisson Becker to the club as well.