The Italy international scored the winner in a 2-1 victory over Newcastle United at the Stadium of Light in October.
However, Borini has found his first-team opportunities limited since his season-long loan switch to the Black Cats.
The striker’s agent had hinted that the Azzurri forward could return to Liverpool in January, but Poyet is keen to keep Borini.
“I would like to talk to him, not agents,” said Poyet.
“He is a very important player for us and he has been unlucky because, if there is one player who should have got a bit more from the manager, it is Fabio.
“I’ve only been here a month and a half so not everything needs to be decided now.
“If I was him I’d be moaning, but he has got great character and we have got a very good relationship.
“Every player wants to play all the time and I’m sure his family are thinking, ‘why doesn’t the manager play him?’ He has done well and is probably the one who has been really unlucky with the situation in games.”
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