Three Lions boss Roy Hodgson revealed after Tuesday's 1-0 win over Estonia that the 19-year-old told him he was "too tired" to start the game, then questioned Liverpool's recovery measures.
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"At no point did Raheem Sterling say he didn't want to play for England," Rodgers told The Independent.
"The boy is being hung out to dry and I dare say the criticism will continue for a few more weeks to come because of this."
He continued: "I'm fed up reading about this club v country row, claims we intervened and put pressure on Roy Hodgson.
"I've read we sent dossiers to the [Football Association] on Raheem Sterling and Daniel Sturridge and I'm having showdown talks with Roy Hodgson on Sunday and all sorts of rubbish.
"The decision not to play him was a managerial decision. I haven't said a single word but have to say I have never seen such rubbish written over the last few days."
Rodgers, though, revealed Sterling has voiced similars concerns regarding his fitness, with the winger rested for the 1-0 home loss to Aston Villa because of it.
He added: "Let me tell you, there have been at least five occasions in the past when Raheem has said he felt tired before games but he's never refused to play.
"We have taken on board what he has said and taken a decision accordingly. I rested him against Aston Villa and we lost the game. But it was our decision to do so – and our decision alone.
"Last season he said he felt tired before the Manchester City game. We won 3-2 and he put in a man-of-the-match performance.
"The point I'm making is you have to take on board advice from your own people and make your own decision, right or wrong."