President Trump stoked controversy on Monday when he said he would meet with the supreme leader if it is “under the right circumstances”.
"If it would be appropriate for me to meet with him, I would absolutely, I would be honoured to do it," the President said in an interview with Bloomberg News. In a separate interview shortly beforehand he referred to Mr Kim as a “smart cookie” for being able to hold on to power in his country.
Albright, the first woman to become the US Secretary of State, said "honoured" was an inappropriate word to use to refer to a meeting with Mr Kim.
"A president doesn't go to a country without any preparation, and 'honoured' would definitely be the wrong way to discuss somebody who is keeping his people in poverty and starving and control," Albright said on CNN.
The Democrat argued the billionaire property developer was overly keen to develop personal relationships with world leaders without acknowledging the concrete political consequences they could have.
"I think that part of the issue is that President Trump seems to believe that he can have just one-on-one relationships. And maybe that's possible in business, but that is not something that is possible as president of the United States," she said. "This is not a matter of charming, by saying you're 'honoured' and he's a 'smart cookie.' "
Albright is one of the most high-profile American diplomats to have visited North Korea on official business and met with the former North Korean leader Kim Jong Il in Pyongyang in 2000. Nevertheless, no American President has yet to meet with the leader of North Korea during their tenure and the notion remains deeply controversial.
Tensions between the US and North Korea have drastically escalated in recent weeks as American and other intelligence agencies have suggested the country was preparing for a possible nuclear test. The Trump administration has said all options, including a military strike, are on the table.
White House Press Secretary Sean Spicer responded to Mr Trump’s bold claims about potentially meeting Mr Kim during a daily briefing, saying the conditions for a meeting between the pair “were not there yet”.
Mr Trump first offered to meet the North Korean leader during the election campaign last June in an attempt to emphasise the differences between him and his Democrat rival Hillary Clinton.
“What the hell is wrong with speaking? And you know what? It’s called opening a dialogue. It’s opening a dialogue,” he said at the time. “If he came here, I’d accept him, but I wouldn’t give him a state dinner like we do for China and all these other people that rip us off when we give them these big state dinners.”
He suggested he would serve Mr Kim and other visiting leaders “a hamburger on a conference table”.