The former Bournemouth boss joined the Reds this summer and, as reported, he has been keen for Elphick and Cooper to join him, along with analyst Tom Webber.
The trio have now joined the Merseyside club, and it is understood that Bournemouth have been paid compensation for their departures and leave on good terms.
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Elphick made 145 appearances for the Cherries as a player, after he joined from Brighton in 2012.
He played for the club in League One, the Championship and the Premier League, and skippered the side before departing for Aston Villa in 2016.
FLASHBACK: Shaun Cooper, pictured playing for Cherries against Nottingham Forest in 2008
Cooper, meanwhile, played 240 times for the Cherries after he joined in 2005 from Portsmouth, before he left for Crawley in 2012 and he was a hero of the club’s greatest escape.
He captained the team as they overcame a 17-point deduction to stay in the Football League in the 2008/09 season.
Both players coached first in the club’s development squad, before progressing into the first team backroom staff under Gary O’Neil.
Iraola inherited both and they played an important role in the Basque boss’s three years at the club.
It is understood that the Cherries are actively recruiting replacements, and it is hoped that they will be in place by the return of the Bournemouth squad on July 11.