Kim Yeon-chul said economic leverage should be used to encourage denuclearization progress.
Senator Chris Murphy, a Connecticut Democrat, told The Atlantic the president "went for the whole enchilada and lost badly."
National Security Adviser John Bolton insisted President Donald Trump's Hanoi visit was somehow a "success" despite acknowledging that no denuclearization agreement was made and North Korea remains a threat to global stability.
The failure of the Hanoi summit made clear the limitations of bilateral talks, especially those between two leaders known to be temperamental.
Explaining the failure to reach an agreement with North Korean leader Kim Jong Un, President Donald Trump told reporters, "Sometimes you have to walk."
President Donald Trump will go ahead with the Wednesday summit in Vietnam, despite being warned by White House aides that the North Korean leader could be mentally unstable.
Donald Trump and North Korea's Kim Jong Un will meet in Vietnam next week for a second summit to discuss possible denuclearization of the Korea Peninsula.
North Korea will "wait with patience and in good faith and, together with the U.S., advance step by step toward the goal to be reached by the two countries," Kim Jong Un said.
A video posted to Twitter by the state department on Sunday showed Pompeo walking alongside Kim in Pyongyang. "Everything is great," Pompeo says.
"We will both prove everyone wrong!" Trump said of himself and Kim Jong Un on Twitter. "There is nothing like good dialogue from two people that like each other! Much better than before I took office."
"If both sides have the political willingness, it is reasonable for them to end the technical state of war on the peninsula first."
"Contrary to our military's suspension of joint Korea-U.S. military exercises, North Korea is conducting military exercises on a scale similar to previous years."
The news comes during a period of heightened tensions between Russia and NATO. Moscow's neighbors have expressed concern about Russia in the wake of the 2014 annexation of Ukraine's Crimean peninsula, as well as Moscow's support for pro-Russian separatists fighting in Ukraine.
"For [Kim], nuclear weapons are a guarantee of his safety."
The two leaders will have a private chat for two hours.
Alongside Japanese Prime Minister Shinzo Abe, President Trump remarked on his upcoming summit with North Korean leader Kim Jong Un.
"We will definitely make sure he gets the best room the hotel has to offer," the CEO of HotelPlanner.com said of his hotel offer to Kim.
"We thank our great Singaporean hosts for their hospitality," White House press secretary Sarah Sanders tweeted.