Palmer, left out of England's World Cup squad this year after a persistent groin injury affected his form, was called up by coach Thomas Tuchel for Nations League fixtures against Spain, the Czech Republic and Croatia in September and October.
However, he was forced to pull out due to a recurrence of the issue, with Tuchel saying the 24-year-old "misses too many opportunities" and then leaving him out of the squad for this month's games.
Asked about Palmer's exclusion after England's 2-1 win over Croatia at Wembley, goalscorer Rogers told the BBC: "He wants to be here. He wanted to get called up and he wanted to come and represent his country as he always has done.
"He was in quite a low place with his injuries last year and didn't feel 100% himself on the pitch, so he wants to be here and he wants to be the best version of himself.
"But he also has to look at his body. He wants to play for 10 or 15 years, he also has to manage himself. 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 a nonsense really that people are questioning someone that wants to play for England. It's a lie."
In his post-match press conference, Tuchel told reporters it was "totally fine" for Rogers to defend Palmer.
"It's his teammate and friend," the German manager added.
"I was very happy with the emotional commitment and physical commitment of Morgan. He was our standout player today."
England next travel to Madrid to face world champions Spain on Sunday in their final Nations League group stage game.
