Hodgson and Rodgers were involved in a dispute in October, arguing over whether Liverpool attacker Sterling asked not to play against Estonia in a Euro 2016 qualifier.
Liverpool manager Rodgers defended his teenage star, and said Hodgson had opted not to start the "tired" Sterling.
Speaking after naming his squad on Thursday, Hodgson said Rodgers was wrong to question his version of events.
"In actual fact, Brendan is wrong to do that," Hodgson said.
"Unfortunately I have 30 witnesses to that extent and Brendan was not there. He got the wrong end of the stick and I'd be happy to put him right. I'd be happy to talk to him about it.
"If a player came up to me one day before the game, when I was intending to play him, just before the session started, and said to me, 'I don't think I can train today because I'm feeling too tired', then I'd definitely leave him out."
Hodgson insisted he never said Sterling had told him he was too tired to play, but rather unable to train.
"I never said that Raheem said he was too tired to play," Hodgson said.
"He did play [coming on in Tallinn] and got us the free-kick which won the game.
"I said he said he was too tired before training and as a result I left him out of the team. You better ask Brendan."
Hodgson said his relationships with both Rodgers and Sterling were "fine", but admitted he felt sad and sorry for the latter.
