The 21-year-old has previously worked with Reds boss Brendan Rodgers first at Chelsea and then when he went on loan to Swansea in the 2010/11 season.
Borini has now arrived at the club's Melwood training ground to have his medical, Press Association Sport understands, and the deal should be completed, barring any hitches, in the next couple of days.
Borini's fee has not been disclosed but it is not believed to be as high as the £12million being reported in some quarters.
Borini scored six goals in nine npower Championship matches for Swansea in their promotion-winning campaign having only joined in March 2011 and he won the penalty in the Wembley play-off final which secured a 4-2 lead and victory over Reading.
With his contract expiring at Stamford Bridge he signed for Italian side Parma, with Chelsea receiving a small sum in compensation, but was immediately loaned out to Roma.
He scored nine times in 24 matches and, after Roma bought half of his rights in January, the Serie A club secured full ownership, at a total cost of around £6million, last month.
Borini's form earned him a place in Italy's Euro 2012 squad but he remained on the bench throughout his country's run to the final, where they were beaten by Spain.
Rodgers knew he needed to bring in more attacking players when he took over from Kenny Dalglish last month.
The Reds' record in front of goal was poor last season, with their 47 goals in 38 matches worse than relegated Blackburn.
Dirk Kuyt has been sold to Fenerbahce while Craig Bellamy, who turned 33 today [Friday], has only a year left on his contract and is attracting interest from hometown club Cardiff.
There has also been plenty of speculation about the future of £35million club record signing Andy Carroll after he was initially linked with AC Milan with the likes of Premier League rivals West Ham and Fulham also coming into the frame.