The 22-year-old has struggled to make an impact at Anfield, especially since the arrival of Rodgers as the new manager.
It is understood that midfielder was offered to Fulham as part of a deal to bring Clint Dempsey to Merseyside in the summer.
Henderson told Sky Sports News: "I love playing for Liverpool, it is a real passion of mine.
"When I put the shirt on it is a real honour to go out and play for Liverpool, so I want to do that for the rest of my career.
"I will keep fighting and I have belief in my own ability that I can be a success here. I have got a good relationship with the manger, I think everybody has, and I feel he has improved my game since he came to the club."
Henderson was signed by Kenny Dalglish for the Reds in 2011.