Former MK Dons midfielder Keith Andrews is to take up a full-time coaching role with the club this season.
Midfielder Gary Deegan has signed a new one-year deal at Southend.
The 23-year-old spent last season on loan at the Stadium of Light from Liverpool, scoring 10 goals as Poyet's team reached the Capital One Cup final and defied the odds to stay in the Premier League.
Sky Sports sources understand the fee is in the region of £14m, and it is now up to Borini to decide whether to move on or stay and fight for his place at Anfield.
"We've agreed terms with Liverpool and we are waiting for the documents to be signed and then we'll move on from there," Poyet told Sky Sports News.
"Like any other deal you do, first you need to sort out the clubs and then you start working with the player
"I'm hoping we can have him because he knows how much it would mean for us to have a player like Fabio."
Liverpool did pick Borini, who was Brendan Rodgers' first signing as Liverpool manager, to start Saturday's pre-season friendly victory over Preston at Deepdale.
But Borini scored just two goals in 20 appearances during an injury-plagued first year at Anfield.
The former Chelsea youth was signed from Parma in the summer of 2012, having impressed the previous season during a loan stint at Roma, where he scored nine goals in 24 appearances.