The Italian, 23, helped the Wearside club preserve their Premier League status while on loan at the Stadium of Light in 2013-14, making 32 league appearances and scoring seven goals.
Poyet confirmed he was able to negotiate with the Anfield giants to now pursue personal terms with Italy international Borini.
"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."
Borini has spent the pre-season under the tutelage of Brendan Rodgers, who selected him to start in a friendly with Preston on Saturday.
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