The 21-year-old Italy international has passed a medical and agreed personal terms at the club's Melwood training ground.
"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 manager Brendan 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."