The Wearsiders claimed that they had agreed a fee of £14million for the 23-year-old, who scored 10 goals while on loan with them last season.
But Borini is with the Reds on their pre-season tour of the United States and is thought to be mulling over his options.
"It's up to Fabio. We have been clear, everything is fine," said Borini. "There's no doubt about his connection with our club, and me and him get along fine, so there is no reason for it not to happen.
"I would love to know how long it will take to sort out, but all I can tell you is that things are changing every day.
"We have targets that we are looking for, we have other options, but we have an agreement with Liverpool and we will do everything possible to get Fabio to the club first.
"If Fabio does not come, it does not set us back, we would move on - we are not a one-player team."
A move to bring defender Marcos Alonso back following his 19-game loan spell in the second half of last season is, according to Poyet, "close, but not close enough."
Poyet has added Jordi Gomez, Costel Pantilimon and Billy Jones to his squad so far this summer, but had hoped to have brought more players in ahead of a pre-season trip overseas.
"I don't think the word is 'frustration', I'm a normal manager that wants his players as soon as possible," the Uruguayan added.
"There are things that are very, very close but we cannot get it over the line so that brings a little nerves because you want the players.
"The idea was to take most of the squad to Portugal on Friday, but it's not going to be possible, and it changes the plans because it's going to be a totally different preparation. Maybe we'll be surprised and sign three or four players on Wednesday or Thursday."