The paper, which broke the story of the latest royal misadventure, says Charles also feels William must share the blame, since he was in the costume shop when Harry picked the offensive outfit, and should have stopped his younger brother.
The visit, according to the Sun, will be private.
Without commenting on the accuracy of the Sun report, Ingrid Seward, editor of Majesty magazine, told The Early Show co-anchor Harry Smith, "There's no way Harry would go to Auschwitz. And I'm sure he…knows all about the Holocaust."
What would have prompted such an insensitive act?
"I'm afraid he's not intellectually very bright," Seward says. "He's rather inclined to do things on the spur of the moment and think about the consequences afterwards, and that's always been his way. But he did have an education at the most expensive and probably the best...school in the world. He had five years at Eton, where he certainly would have learned about the Holocaust. I suppose you can put it down to gross stupidity.
"I'm totally in agreement with (the politicians) who said he should come on television and (apologize) himself. I think both William and Harry, when they get into scrapes, should come out and be their own masters. (If he had apologized) in person, I think it would have died down."
Ken Whafe, who was a bodyguard for Princess Diana and sons William and Harry, wondered why Harry's current bodyguards didn't warn him of the impropriety of what he was about to do. He told Smith royal bodyguards will often discreetly offer such advice.
He adds that he agrees with Seward: "Harry did something that was stupid, and I think there was no sinister motive for him to offend anybody."
CBS News Correspondent Mark Phillips, in London, calls Harry "the current royal gaffe champion -- accused of binge-drinking and of attacking a prying photographer last summer."
"As royal gaffes go," notes Phillips, "this one has set new standards for effectiveness. It's managed to upset everybody... Holocaust survivors and victims families, ethnic and religious groups, war veterans and those who lost people in the war.
"Some are now wondering if Harry's desire to pursue a military career is really a good idea. Apology or not, this is an image that may haunt him for the rest of his life."
Before the report of Charles' demand came out, the Simon Wiesenthal Center urged Harry to make just such a trip. "We strongly urge Prince Harry to accompany the British delegation on Jan. 27 to the Auschwitz death camp to commemorate 60 years since liberation," said Rabbi Marvin Hier, dean and founder of the Wiesenthal center. "There, he will see the results of the hated symbol he so foolishly and brazenly chose to wear."
"What Harry did was both stupid and evil," said Lord Janner, a senior figure in Britain's Jewish community. "The time has come for him to make a public apology."
Harry made his first apology Wednesday night, in a statement, just as The Sun newspaper's first edition was hitting the streets with a big headline saying HARRY THE NAZI and a picture of the young royal wearing a swastika armband.
"I think a lot of people will be disappointed to see that photograph and it will cause a lot of offense," said Michael Howard, leader of Britain's main opposition Conservative Party.
"I think it might be appropriate for him to tell us himself just how contrite he now is," added Howard, who is Jewish.
Prime Minister Tony Blair sidestepped the furor. "Prince Harry has made it clear that he is very sorry about it. The rest of it is best to leave Buckingham Palace to comment on," he told BBC radio.
Doug Henderson, a legislator and former defense ministry official, said Harry was unfit to take his place at Sandhurst, the military academy.
But Janner said: "I would send him in the army as fast as possible. I hope that would teach him not to behave like that."
The action also upset many Israelis.
Vice Premier Shimon Peres called it "very bizarre."
"Well, the next time he will dress up and behave like a prince," Peres said.
The young prince has been a favorite topic of the tabloids in the past few years, reporting on, alleged misadventures and scrapes with propriety, including a last October and who claims she was ordered to help him cheat his way through an important examination.
And in 2002, Charles made Harry spend a day at a drug rehabilitation center after he was caught smoking marijuana and indulging in underage drinking.
Harry is currently enrolled at Sandhurst, where he is.
He is third in line for the throne.