Midfielder Downing - a £20 million signing from Aston Villa under Kenny Dalglish - continues to be unhappy about comments Rodgers made earlier in the season apparently questioning his commitment to the Anfield club.
"I was obviously upset. I always try to give my best," said the England player in Sunday newspapers. "You’ll have to ask the manager what he meant by his quotes. I would have preferred it obviously if it was private."
Rodgers apparently suggested Downing and some of his team-mates were "lazy" after the 3-2 defeat to Udinese in the Europa League on Thursday.
"Lots of others in the team are putting their bodies on the line and the ones that don’t stand out like a sore thumb," said Rodgers.
"Bravery is a lot of things," commented Downing in response to the Rodgers claims. "If you want tackling, running around bustling, that obviously is not my game. But bravery is also taking the ball when you are losing a game and creating things, which I try to do.
"We’ve had a few conversations. You can sit in offices for hours and talk about my game, but the only way I can prove it is out on the field. When I’m given that chance I have to take it. I thought I did OK against Udinese.
"I’ve three years left on my contract so I’m going nowhere. I will fight," added Downing, who is also vowing to return to the England side. "I’ve worked hard to get here so I don’t want to leave in a rush.
"I don’t think any England manager is going to pick me if I’m sat on the sidelines. That’s obvious. But I’ll never give up on England. If I have a good season here, hopefully Roy Hodgson will put me back in."