Downing admits he is disappointed at not featuring as a first-team regular so far this season.
He told liverpoolfc.com: "I don't want to leave and I don't want to play for anyone else. I'm enjoying it here. Yes, I'm disappointed that I'm not in the team at the moment but football is a funny game.
"I'll fight for my place here. I have done that since I was 10 years old when I was trying to get through at Middlesbrough."