Sky Sports reports that the pair have agreed terminations of their Bournemouth contracts and are expected to finalise their deals with Liverpool this weekend.
Bournemouth had hoped both coaches would remain part of the club's setup under new manager Marco Rose, but that will no longer be the case. Rose is now set to bring in his own assistants, with the pair's departures already factored into the club's planning for next season.
Elphick, 38, is a former player at Brighton, Bournemouth and Aston Villa who worked closely alongside Iraola at the Vitality Stadium. He is expected to serve as assistant head coach at Anfield. Elphick had also been approached by Bristol City, who considered him for their head coach vacancy, but turned the opportunity down in favour of rejoining Iraola, underlining his preference to remain within the top-flight coaching structure.
Cooper, 42, came through Bournemouth's ranks as a player, making 240 appearances for the Cherries after signing in 2005. He retired from playing in 2017 and moved into coaching, first with the club's youth teams and then within the senior setup. Credited as a set-piece specialist, his arrival at Anfield addresses an area where Liverpool struggled for large portions of last season.
The pair's arrival completes the core of Iraola's new backroom team ahead of pre-season preparations.
Liverpool's appointment of Iraola marked a significant shift in direction at Anfield following the departure of Arne Slot, and the retention of trusted lieutenants from the Spaniard's Bournemouth setup suggests a clear continuity of approach is planned for next season.