The Black Cats are believed to have agreed a £14 million deal for the Italian, but the striker has spoken of his desire to stay and fight for his place at Anfield
Fabio Borini has flown with Liverpool for their pre-season tour of the United States, despite being on the verge of a move to Sunderland.
The Italian, 23 helped the Wearside club preserve their Premier League status while on loan at the Stadium of Light in 2013-14, making 32 league appearances and scoring seven goals.
Sunderland head coach Gus Poyet revealed on Saturday that a permanent deal for the forward was close to completion, after the two clubs agreed a fee - reported to be in the region of £14 million.
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Poyet told Sky Sports News: "We've agreed terms with Liverpool and we are waiting for the documents to be signed and then we'll move on from there.
"Like any other deal you do, first you need to sort out the clubs and then you start working with the player.
"I'm hoping we can have him because he knows how much it would mean for us to have a player like Fabio."
However, with personal terms still to be agreed - and amid reports that the player may be reluctant to return to the Stadium of Light - Borini has now flown out to the US with the rest of the Liverpool first-team squad.
Borini had previously hinted at his desire to make an impact at Liverpool as they prepare themselves for the return of Champions League football next season.
He told reporters in May: "I have enjoyed every second, not every minute, of it here [at Sunderland]. It has been great from the fans here towards me, the city and everyone in the club.
"The love and friendship that they give me through the season has been incredible.
"Liverpool do need a bigger squad next season because you can't play four competitions with the same 11 players.
"But it is now down to the Liverpool point of view and their sporting director. It is not down to my words.
"It depends on what Liverpool say. I can say nothing now because anything is possible."
Brendan Rodgers' side will play a friendly against Roma in Boston on Wednesday before competing in the International Champions Cup in a group that also contains Olympiacos, Manchester City and AC Milan.