The 23-year-old turned down moves to both clubs, and is ready to prove to Brendan Rodgers that he still has a place on Merseyside.
"The transfer market was like a storm," Borini told gianlucadimarzio.com. "Now it has passed, I don't have problems anymore. I'm included in the Champions League's list and we are training well. In the summer, there was a lot of talk, but there weren't big misunderstandings.
"My wish is to play in a top club and Liverpool are a top club. There have never been real problems, I know what my goals are and how I can reach them. Staying at Liverpool was one of them. I'm in a great team that tries to win every competition and that's what I want.
"If I stayed here, it's because I thought I can play. It's not out of thirst of greatness or to keep my spot here. I'm not like that. If I stayed at Liverpool it's because I think I could compete."
Liverpool resume their Premier League season when they welcome Aston Villa to Anfield on Saturday.