A spokesman for Clarence House said: “We do not comment on private conversations.
“But we would like to stress that the Prince of Wales would not seek to make a public political statement during a private conversation.”
Charles was being shown around the Museum of Immigration in Halifax, Nova Scotia, along with Camilla, the Duchess of Cornwall.
The royal couple paid tribute to World War Two veterans and their families, and during the course of the visit they spoke to museum volunteer Marianne Ferguson.
Ms Ferguson told the Prince she fled to Canada with her family in 1939, not long before Hitler annexed the Baltic coastal Free City of Gdansk.
After meeting Charles, the 78-year-old told : “The Prince said ‘And now Putin is doing just about the same as Hitler’.
“I must say that I agree with him and am sure a lot of people do.
“But I was very surprised that he made the comment as I know they [members of the Royal Family] aren’t meant to say these things.
“I told the Prince that while my family and I were lucky to get a permit to travel, many members of my relatives had permits but were unable to get out before the war broke out on September 1.
“They were sent to the concentration camps and died.”
The Mail reported that the Prince made his comments while surrounded by media and they were heard by several witnesses.
Mr Putin’s press secretary Dmitry Peskov told : “I don’t know anything about it. I can’t really trust the Daily Mail as a source.”
Deputy Prime Minister Nick Clegg backed Prince Charles on his LBC Radio phone-in, saying: “Prince Charles should be perfectly entitled to express views in the confidence that he’s expressing them privately.”
But Labour (Co-op) MP Mike Gapes, who represents Ilford South, said the Prince “should abdicate” if he wants to make controversial statements.
He said on Twitter: “If Prince Charles wants to make controversial statements on national or international issues he should abdicate and stand for election.
“In constitutional monarchy, policy and diplomacy should be conducted by parliament and government. Monarchy should be seen and not heard.”
UKIP leader Nigel Farage, who has backed Mr Putin’s anti-EU stance in the past, said: “Prince Charles has made those comments – I know some people feel that way about Putin.
“I think there’s a difference. The difference is right from the very start Hitler was expansionist, and we haven’t see very much evidence of that until now from Putin and arguably, what’s happened in the Ukraine is because he’s been poked with a stick by the rest of the world.”
Charles and the Russian leader are due to meet next month when they attend the 70th anniversary of the D-Day landings in Normandy on June 6.
Mr Putin has faced international anger over Russia’s actions in neighbouring Ukraine, including the controversial annexation of Crimea.
In March, former US Secretary of State Hillary Clinton reportedly said Mr Putin’s claim to be be defending ethnic Russians in Crimea was “what Hitler did back in the ’30s”.
She later backtracked, claiming she was not making a direct comparison but that Russia’s behaviour was “reminiscent” of Germany in the build-up to the Second World War.
Prince Charles has been known for speaking his mind on issues such as architecture and the environment, but he rarely makes his feelings known on diplomatic matters.
There is an ongoing legal battle over the publication of letters he has sent to politicians, with the attorney general concerned their release could compromise the Prince’s neutrality and create constitutional problems.