Rodgers openly questioned Downing's performances to the media earlier this weekend, and challenged the England international to produce the goods in a Reds shirt.
Downing told reporters that he believes Liverpool boss Rodgers should have kept his thoughts private between the pair.
He said: "I was upset and you'll have to ask the manager what he meant with that quote. I always try my best.
"Bravery is a lot of things. If you want tackling and running around bustling, that's not my game. But bravery is also taking the ball when you're losing and creating things. That's what I try to do.
"I'm disappointed. Yet if that's how the manager sees it, there's not a lot I can do about it.
"I just have to keep working hard and do what he tells me to do. I have to perform on the field and get my head down."