Borini spent the 2013-14 campaign on loan at the Stadium of Light and scored seven goals in 32 Premier League appearances.
Head coach Poyet made no secret of his desire to bring the 23-year-old to Wearside on a permanent basis in the last transfer window, but an expected move collapsed.
The forward's prospects of regular first-team football at Anfield appear slim, though, with Liverpool manager Brendan Rodgers leaving the forward out of his squad for Saturday's 0-0 draw against Poyet's side.
And Poyet is keen to strike a deal in January if the opportunity arises.
"I would love to [sign Borini]," he said. "It is difficult to explain [why the previous move collapsed].
"It was a long summer and the things everyone knows, we agreed a price and we flew to America [for Liverpool's pre-season tour] and he didn't come.
"The bottom line is he is not with us. I am trying to concentrate on what we have got."
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