Sterling, 19, started in England's 5-0 victory over San Marino at Wembley the previous Thursday but was left out of the starting line-up in Tallinn after telling Hodgson he felt tired, although he came off the bench to win the free-kick from which Wayne Rooney scored the only goal.
Rodgers insists that Sterling never said he did not want to play and that it was Hodgson's decision to leave the forward out of the starting XI.
"I'm fed up reading about this club v country, claims we intervened and put pressure on Roy Hodgson," Rodgers told The Independent.
"I've read we sent dossiers to the FA 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 past few days."
Rodgers feels Sterling is being unfairly treated and does not believe the criticism will end in the near future.
"At no point did Raheem Sterling say he didn't want to play for England," he added.
"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."
Sterling has made 10 appearances for Liverpool in all competitions so far this season - scoring three goals - whilst also making four outings for England.
Rodgers admitted that Sterling had previously expressed concerns about fatigue, but insisted the club always made the final call on whether he would start a particular match.
"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," Rodgers continued.
"We have taken on board what he has said and taken a decision accordingly. I rested him against Aston Villa (13 September) 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 (13 April). 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."
