The winger has failed to make an impact on Merseyside following his £20m move from Aston Villa last summer and has now found himself behind Raheem Sterling and Fabio Borini in the pecking order at Anfield.
"It hasn't quite worked out for him as he would want it. The big challenge for him now is that commitment to the cause to fight, because he has the qualities," Rodgers told reporters.
"You have to work hard, you have to fight for the shirt. We have seen that in a number of the young players and if you don't do that then longer-term you won't be here. It is as simple as that really."
Downing has scored three goals in 55 appearances for Liverpool.