Despite having agreed a fee for the Italian forward last month, a deal for the 23-year-old has yet to be completed, but the Black Cats boss remains confident and is urging patience
The Italian frontman spent last season on loan with the Black Cats and the club's Uruguayan boss confirmed in July that a bid, reported to be £12 million, had been accepted by the Reds.
Sunderland, though, have as yet been unable to finalise the deal, with suggestions that Borini is keen to remain at Anfield, but Poyet insists the former Chelsea and Roma forward is eager to complete a permanent switch.
"I cannot tell you everything because it's not fair on anybody," he told reporters. "But it's not like Borini doesn't want to come and we're trying to convince him.
"I talked to Fabio and that is not the case. It's about doing it at the right time.
"He's come back to fitness and training with the team. So he wouldn't be fit for tomorrow anyway. So calm, please!"
Poyet admits the club need to bolster their attack and will move on to other targets if their pursuit of Borini continues to stall, though the 23-year-old is nevertheless their top target.
"I'm just not trying to put a deadline on it," he added. "We need a player in that position and we are ready to move on and have an option.
"But of course, what happens if you move on and he's still available? It's a tricky situation. But it's impossible for me to say Monday, Wednesday or the 31st [of August]."