News broke on Friday evening that the Black Cats had agreed a £14 million price with the Reds for their Italian striker.
However, Borini has since jetted off to the United States with the Anfield side for their pre-season tour, which led to suggestions that the former Swansea City man still had a future with Brendan Rodgers' team.
Borini, who spent last season on loan with Sunderland, has been a primary target of Poyet all summer, and the Uruguayan coach claimed that a deal was still very much in the pipeline.
The former Brighton boss also slammed the media for stories that suggested the signing was nowhere near its completion.
Speaking to the Chronicle, Poyet said: "I can assure you though that there is no disappointment on my part that it has not happened yet. It is normal. And with all respect to the media, they do not know everything that is going on. It is not for me to explain but it is not just a question of Fabio wanting to come or not.
“There are many, many things to look at and unfortunately I cannot comment on those. We are involved in negotiations and you never know how long that can take. There’s nothing new to report as things stand but we are still hoping to close the deal completely.
“We have been clear, everything is fine, 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. It's up to Fabio.
“If Fabio does not come, it does not set us back, we would move on - we are not a one-player team. But I hope the fact that we are after players like him shows the quality of players that we are trying to bring to the club."
Borini was signed from Roma in the summer of 2012 for around £11 million but the striker struggled to find form before he was injured on international duty with Italy Under-21s in October that year.
The Italian then saw his season rocked with another setback when he suffered a dislocated shoulder in the 5-0 win over Swansea City in February 2013.
However, last season proved to be much more fruitful for Borini, as he netted 10 goals for the Black Cats, including the opener in their League Cup final defeat to Manchester City.