Borini's future has been uncertain for much of the summer, as Sunderland wish to make their loan deal for the striker last season a permanent one.
Borini had a successful loan spell at the Stadium of Light last season as he scored key goals in their campaign to stay in the Premier League, and during their run to the League Cup final.
Liverpool had accepted a £14 million bid for the Italian forward a couple of weeks ago, but the striker seems determined to fight for his place at Anfield, and wants to play Champions League football.
The report also highlights the possibility of a second season-long loan spell being mooted for Borini.
The Italian has a big choice to make, and given what has occurred over the summer and during pre-season at Anfield, any notion that he may have his first-team chances at Anfield severely restricted may not be the case.
Luis Suarez's departure has left a void, but a hamstring problem suffered by Daniel Sturridge in the United States - combined with his injury record - is likely to concern those at Anfield. As a result more striking options are needed at the club, and Borini could prove to be valuable to Liverpool's cause in the coming season.
The ball is firmly in the Italian's court given the offer was accepted by Liverpool a matter of weeks ago. His decision could have a big say on Liverpool's transfer business for the remainder of the window.
Liverpool manager Brendan Rodgers has said it is crucial for him to land another striker, and the departure of Borini could mean he'll target two more.
As for Sunderland, if they land Borini, it will mean that one of Gus Poyet's key targets has been secured ahead of the new Premier League campaign.