The 19-year-old midfielder was omitted from the Three Lions side ahead of the Euro 2016 qualifier last weekend, before coming off the bench to make an instant impact in Tallinn.
Hodgson revealed that Sterling was left out due to the player "feeling tired", which led to suggestions that Rodgers had a major say in the team selection.
"I'm fed up reading about this club v country, claims we intervened and put pressure on Roy Hodgson," the Reds boss told The Independent.
"I've read we sent a dossier 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.
"At no point did Raheem Sterling say he didn't want to play for England. 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."
Liverpool are back in Premier League action this weekend when they host Queens Park Rangers.