The Italian forward spent last season on loan at The Stadium of Light - scoring 10 goals in all competitions for Gus Poyet's side.
Having returned to Anfield at the conclusion of his loan spell, the former Chelsea and Parma man has been a key target for Poyet ahead of his first full season at Sunderland.
However, his agent Marco De Marchi claims the 23-year-old is determined to secure a place in Brendan Rodgers' starting XI.
"Sunderland have made a large offer of €18 million to sign him, but Fabio wants to stay," De Marchi told tuttomercatoweb.com.
"His project with Liverpool started two years ago when he left Roma. He began a journey and it is a major project for him.
"He should not be judged on his first year [at Liverpool].
"He was out for six months with various injuries and during the past year he took a step back by going to Sunderland and he has had an extraordinary season.
"Now, after the loan, Liverpool have not told me anything. I think he will stay, he is convinced that he will play."
