Kim Byeong-uk fled the North 16 years ago and now runs the North Korea Development Institute, an economic think tank in Seoul.
He did not detail the nature of the new US communication with the North.
She also said it was up to the Americans whether they would "meet us at a meeting room or encounter us at nuclear-to-nuclear showdown".
USA defense and intelligence officials have repeatedly assessed the North to be on the threshold of having the capability to strike anywhere in the continental US with a nuclear-tipped missile - a capacity that Trump and other US officials have said they would not tolerate.
The latest talks were held on the northern side of the Panmunjom truce village, between 15:00 and 17:00 local time (06:00 and 08:00 GMT), Mr Moon's office said.
The leaders of North and South Korea met Saturday for a second time, South Korea's presidency announced.
Police respond to call of shots fired at Trump National in Doral
Before Oddi began shooting, he removed a flag from the flagpole and draped it over the counter of the hotel lobby, police said. Senior officers told USA news a "confrontation" with a man at the resort escalated and police shot the man.
President Moon Jae-in called an emergency meeting and government spokesperson Kim Eui-kyeom told reporters, "We are trying to figure out what President Trump's intention is and the exact meaning of it".
"Everybody plays games", Trump declared on Friday.
Also, for talks to commence, North Korea must detail a preliminary roadmap as to what it would offer in terms of denuclearization.
Pyongyang on Friday said in a statement that it is still willing to "sit with the USA side" for talks "at any time". While China may have been wary of the pace of diplomatic activity between the two Koreas and Washington in recent weeks, it certainly does not want a return to the tension of the pre-Olympics period when the Trump administration was threatening military strikes on its doorstep, and China had little choice but to play a supporting role in the US pressure campaign.
The comments reflected a broadly shared perception inside the White House and State Department that the two leaders still want to get together and there will be a meeting eventually.
He added that North Korea released a "very nice statement" in response to his canceling the summit in Singapore.
Indeed, the idea of peace on a peninsula still technically at war has started to feel so real that there's a boom in property sales inside the demilitarized zone (DMZ).
We must keep eyes on Iran
Facing that reality, Le Maire said the European Union should strengthen its own legal arsenal against any such future USA threats. Le Maire said Paris would seek to strengthen Europe's ability to block sanctions and provide investment finance to companies.
Trump's letter issued a veiled threat to Kim, saying: 'You talk about your nuclear capabilities, but ours are so massive and powerful that I pray they will never be used'. "We'll see what happens".
North Korea's Vice-Foreign Minister Kim Kye-gwan said Trump's cancellation decision was "extremely regrettable".
"It's a fortunate thing that the possibility of the North Korea-U.S. dialogue is still alive without being terminated", the South's Yonhap news agency quoted spokesman Kim Eui-kyeom as saying.
What about Mr Trump's cancellation letter?
Saturday's meeting at the DMZ follows a daylong summit last month in which Moon and Kim embraced, planted a tree and talked alone for more than 30 minutes.
The United States and North Korea have yet to call off a weekend meeting in Singapore between senior officials on both sides, despite uncertainties surrounding President Donald Trump's decision to cancel his summit with Kim Jong Un. There, former leader Colonel Gaddafi gave up his nascent nuclear programme only for him to be killed by Western-backed rebels a few years later.
Weinstein accusers Rose McGowan, Asia Argento respond to arrest: ‘We got you’
The criminal sex act charge relates to a 2004 meeting between Weinstein and Lucia Evans , who was a new, young actor at the time . Asked about the raft of other allegations against Weinstein, Mr Brafman said the case was a question of crime, not bad behavior.