The England international winger has not lived up to his £20 million price tag following his move to Anfield from Aston Villa in the summer of 2011.
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Rodgers has challenged Downing to show the passion required to play for Liverpool or face being shipped out of the club.
"Stewart is a good guy and a good lad who came here for an awful lot of money," Rodgers told The Guardian.
"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. He has a wonderful left foot but what is important is that it is more than talent.
"Talent alone is not enough. 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."
Rodgers insists the responsibility over Downing's performances rest solely on the player's own shoulders.
He added: "We are coaches, not magicians. For me football players are no different to plumbers, joiners or bricklayers. They are self-employed and us as medics, coaches and managers are the tools that help them to be better.
"We will give them everything to help them to be better, but if they don't want to show that self-motivation, and if you haven't got that desire and hunger to succeed, then it can be very difficult for you."Jose Enrique's Liverpool career could also be under threat, Rodgers revealed. Despite the former Newcastle man enjoying a relatively promising start under Kenny Dalglish last season, his form has dropped recently following a spell out with injury.
Rodgers added: "I don't waste my time waiting until January to tell both players [Downing and Enrique] what I know now. So they know in relation to the demands of what we want and certainly what it is going to take for us to succeed and for them to fit into the dynamics of the group."