Suggestions that Palmer does not want to represent his country have been rejected by Rogers, who says the Chelsea midfielder has been dealing with the effects of injuries and must be allowed to manage his fitness.
“It's a lie. He wants to be here,” Rogers told BBC Radio 5 Live after England's 2-1 Nations League victory over Croatia.
“He wanted to get called up and he wanted to come and represent his country as he always has done. People just see it as news and say the easy thing.
“We all know he had a difficult year last year with injuries and I know personally he's been managing injuries ever since.
“He was in quite a low place with his injuries last year and didn't feel 100 per cent himself on the pitch, so he wants to be here and he wants to be the best version of himself.
“Of course he does, who wouldn't? But he also has to look at his body. He wants to play for 10, 15 years, he also has to manage himself.”
Palmer was left out of Thomas Tuchel's squad for the fixtures against Croatia and Spain after missing previous opportunities to impress the England manager. He had also made himself unavailable for the October international camp.
Tuchel explained that the decision was not a question of Palmer's desire to play for England, but the limited time he had spent with the national team.
“Everyone is desperate to play for England and Cole is no exception,” he said. “It counts when you are in camp and he missed too many chances to impress.”
Rogers, who scored England's opening goal against Croatia before Harry Kane added a penalty, believes his team-mate's commitment should not be questioned.
“If he has to manage himself in the short term for the greater good, then he has to do that and you have to fully support it.
“It has nothing to do with England or not wanting to play with England.
“It's nonsense that people are questioning someone who wants to play for England.
“You see the joy and smile on his face, how much he loves the game, so I honestly don't get it and he doesn't get it.
“But it's something he has to deal with but I can assure you he wants to be here and he wants to play for his country.”
Asked whether he wanted Palmer back in the squad, Rogers was emphatic.
“Of course I do. What a player he is. What a person he is and it's someone you want to see in an England shirt.
“He's got so much quality, so much ability.
“We all know what he can do, what he's done, so of course we want to see him in an England shirt and hopefully sooner rather than later.”
Tuchel was not troubled by Rogers speaking out in support of his club-mate and friend.
“It's totally fine. It's his team-mate and friend, why wouldn't he. I haven't seen it so I wouldn't comment,” the England manager said.