The 21-year-old Italy international passed a medical and agreed personal terms at the club's Melwood training ground earlier on Friday.
Borini moves to Anfield for a fee journalists understand is in the region of £7.9million.
Borini will be reunited with Rodgers, with whom he worked in the Chelsea youth set up, for a second time having spent a short spell on loan at Swansea in the 2010/11 season.
The young forward, who scored six goals in nine matches for the Swans, is pleased to be back in the country and hopes he can help fire the club back into the Champions League.
"I felt really good because I wanted to come back to England one day. I knew sooner or later I would be back," he said.
"Being back with Liverpool is even better because I can show the people what I can do - we've got the Europa League and lots of things to play for.
"I hope to achieve the maximum I can, by scoring goals, getting into the Champions League and all of the things the club also wants - and all the other players.
"It's a top club that has won a lot of trophies in the past - and will do so in the future we hope as well."
Borini believes he will find it easy to fit in at Anfield as he is well aware what Rodgers expects and how he wants to play.
The fact the Reds boss made him his first signing since taking over from Kenny Dalglish last month also means a lot to the Italian.
"I've got the trust of the manager and the club, which for me is very nice and I'm proud to be here," he told liverpoolfc.com.
"He was very important. I wouldn't say it was the key, but most of it came from him.
"Even during last season, I spoke with him a lot. I'd text him and ask if he was okay and congratulated him on the results he got with Swansea.
"First of all it is a relationship of friendship and afterwards it is a working one. That is more important than anything.
"I used to play 4-3-3 with him and he is very good to play football for.
"It is great to play football for him because you can have fun and you can get results.
"I could see at Swansea the people loved him because the football was attractive. People like to see football (played) like this."
Borini only moved back to his homeland last summer as, when his contract at Stamford Bridge expired, he joined Parma who paid a small fee in compensation.
He was immediately loaned out to Roma where he scored nine times in 24 matches and, after they 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.
Watch: Carroll to fight for Liverpool future... More Videos