Borini scored 10 goals in 40 appearances for Sunderland during his season-long loan from Liverpool and played a key role in the club's run to the Capital One Cup final.
The Italy international recently expressed his wish for a chance back at Liverpool and his contribution at the Stadium of Light has been praised by Poyet.
The 23-year-old marked his final appearance for Sunderland with a goal in the 3-1 defeat by Swansea on Sunday and Poyet is now waiting to see what happens regarding Borini.
"Fabio has been a key player for us; he's been outstanding," Poyet told the club's official website. "He's a terrific character and a very good professional.
"I'm not sure what's going to happen but he is very much a player any manager would like in their squad."
Poyet has also hailed the progress of Argentinian defender Santiago Vergini, who made 14 appearances for Sunderland after arriving on loan in the January transfer window.
"Vergini pleases me for a lot of different reasons," added Poyet.
"Firstly because I brought him here and I know him as a player and as a person, and know how important it was for him to succeed in England.
"He was playing out of position so it is even more impressive what he has done."