Borini scored 10 goals in 40 appearances on loan at Sunderland last season and played a key role in the club's run to the Capital One Cup final.
Sunderland agreed a reported £14million fee with Liverpool last month, but have yet to reach an agreement with the once-capped Italy international.
A shoulder injury suffered by Borini during Liverpool's pre-season tour of the United States has held up negotiations, while there have also been reports that the 23-year-old wants to fight for his place at Anfield.
Poyet remains determined to sign Borini and is also targeting further additions to his squad after recruiting six players so far this summer.
"We're working on it," Poyet told Sky Sports News. "It's not easy, it's been a difficult summer, I think, for everybody.
"We are not finished. We need more bodies. We are looking for specific, quality players."
When asked if there is a timescale to conclude the Borini deal, Poyet said: "If you asked me last week, I probably would have said this week, but I cannot tell you next week because we will wait for him.
"Of course, we need to do other business that is important for the team. I have a shape. You don't realise it, but we are 10 days away [from the start of the season]."