The Italian arrived at Anfield in 2009 during Rafa Bentiez's rein as a replacement for Xabi Alonso, but a combination of injuries and a lack of form saw his playing time limited.
Liverpool have sent the 28-year-old on loan to Juventus and AC Milan in the last two years, with the latter opting against making his move permanent.
Aquilani is aiming to play regularly for the Reds next season and is hoping to impress Rodgers during the club's pre-season tour of the United States this month.
"How can you pay £20 million for a player and this player is never taken into consideration?" he told BBC Sport.
"I never asked to leave but I was always pushed by the club. I am here and I would like to play for Liverpool but it doesn't depend only on me."
Aquilani played 45 minutes of the 2-1 defeat to former club AS Roma on Thursday and the midfielder is keen to put his frustrations behind him and make a name for himself at Anfield.
"The situation changed after Benitez left," Aquliani continued.
"In past years, I felt forced out. I never asked to go on loan but they made it clear it was better for me to go.
"I think Liverpool people have to know that I have never said I wanted to leave the city because my family and I were not happy here. This is not true.
"I cannot say I have unfinished business in English football because I really never played but, the little I did, I believe I was able to show to Liverpool fans that I have quality and that I could have helped the team."
