The Italian was the subject of interest from Sunderland throughout the summer, and Queens Park Rangers also made an offer on August 31.
Borini opted to stay on Merseyside and he claimed that he is up for the challenge of forcing his way up the Anfield pecking order.
The Mirror quotes him as saying: "I have committed to the club, and I want to earn a place in the side. It is my aim to return to the team and to score goals for Liverpool. I know it will not be easy, it will be very challenging. But I am motivated.
"Yes, I heard some criticism, but I made the decision and it is my job to prove I can be worth a place. That's what I aim to do now.
"I know that it is a real challenge now to find a space in the Liverpool side, because the competition here is so great, and there are many good players. But for me, that should be a motivation, not a problem."
Borini signed for Liverpool for around £12m back in 2012.