The 26-year-old, who has also been courted by Lazio, reportedly flew to his native Italy on Wednesday in a bid to tie up the deal.
Reports in the Italian media have suggested he will complete a £5.3million permanent transfer, although the finer details of the proposed move are still to be finalised.
A successful conclusion to the negotiations would see Borini leave the Stadium of Light two years after his £10million switch from Liverpool on a four-year deal.
The former Chelsea trainee has made a total of 90 appearances - the first 40 of them on loan - and scored 17 goals, including one in a Capital One Cup final defeat by Manchester City in March 2014, for the Black Cats.
He initially joined Sunderland on a season-long loan deal from Liverpool in September 2013 and played a key role in the club's League Cup campaign, to the extent that then head coach Gus Poyet spent much of the following summer trying in vain to complete a permanent deal for him.
Borini opted instead to return to Anfield and fight for his place, but managed just 18 appearances, 13 of them as a substitute before the Black Cats finally got their man.
By that point, however, Poyet had gone and the former Roma frontman struggled to rediscover the form which had initially made him a fans' favourite under successors Dick Advocaat, Sam Allardyce and David Moyes.
Most of Sunderland players have a relegation clause in their contracts which triggers a 40 per cent reduction in wages, and the club's fall into the Sky Bet Championship at the end of last season means those clauses have now been activated.
The club are currently trying to build for life in the lower division against the backdrop of takeover talks and with Moyes yet to be replaced, and some of its bigger earners are certain to leave with Borini, Lamine Kone, Wahbi Khazri and Jeremain Lens, who spent last season on loan at Turkish side Fenerbahce seemingly heading that list.
Keeper Jordan Pickford has already secured a £30million switch to Everton, while striker Jermain Defoe is expected to leave on a free transfer courtesy of his own relegation clause.
