The Italian striker, 23, was heavily linked with both Sunderland and Queens Park Rangers over the summer, but a deal with either team failed to happen.
Borini is now eager to work hard at Anfield in the coming months and make himself a part of manager Brendan Rodgers's plans.
"This summer there has been many rumours. My wish is to play in a top club and Liverpool are a top club so there have never been any real problems. I know my aims and how I can achieve them," he told Gianlucadunarzio.com.
"Staying at Liverpool is one of them. I'm in a big side who try to win each competition, and that is where I want to be. I stayed because I think I am able to play here."
Borini scored 10 goals in 40 games last season while on loan at Sunderland.