The Italian frontman spent last season on loan at the Stadium of Light and some reports suggest Liverpool have already accepted a £14 million offer for the player.
Poyet has as yet been unable to finalise the deal, but stressed that Borini wanted to return to the club.
"I cannot tell you everything because it's not fair on anybody," he said. "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!"
Despite the delay in bringing Borini to the club, Poyet is not prepared to put a timeframe on the prolonged deal.
"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]."
