Such moves would be created to reinforce to Kim that he faces more potential economic pain, especially if he takes any provocative action after Trump on Thursday abruptly canceled their planned June 12 summit in Singapore.
Trump said they are still considering June 12 in Singapore for the summit with Kim Jong Un.
Trump also told reporters at the White House on Saturday that conversations between Kim and South Korean President Moon Jae-In "have gone very well".
U.S. officials crossed into North Korea on Sunday to hold talks on preparations for a possible summit, a U.S. newspaper reported, as North Korean leader Kim Jong Un reaffirmed his commitment to meet with U.S. President Donald Trump.
South Korean President Moon Jae-in met Kim at the border on Saturday for the second time in a month to discuss how to keep Kim's summit with Trump on a track.
On Thursday, Trump had called off the meeting scheduled for June 12 in Singapore, citing hostile remarks from the North. Pyongyang responded the following day, saying that it is open to meeting with the U.S. at any time, which then prompted Trump to hint at the possibility of the meeting going ahead as planned. "They simply stood us up".
Trump previous year launched what his administration has called a "maximum pressure" campaign against North Korea combining the toughest-ever USA and worldwide economic sanctions with diplomatic actions and the Republican president's military threats and preparations. Moon was accompanied by National Intelligence Service Director Suh Hoon, and Kim Jong-un by Kim Yong-chol, vice chairman of the central committee of the ruling Workers' Party.
The meeting came hours after South Korea expressed relief over revived talks for a summit between Trump and Kim.
Putin made the comments during a joint press conference in St. Petersburg with French President Emmanuel Macron, who is in the country on his first official visit.
The last straw, according to the White House, was an insult aimed at U.S. Vice President Mike Pence earlier Thursday in a statement by North Korea's vice foreign minister, Choe Son Hui.
"We have inwardly highly appreciated President Trump for having made the bold decision, which any other USA presidents dared not, and made efforts for such a crucial event as the summit", Kim Kye Gwan's statement said. Kim and Trump's initial decision to meet followed months of war threats and insults between the leaders over the program. Both Korean leaders say they're invested in making that summit happen.
But even though he was confident of Kim's intentions, Moon admitted that difficulties likely lie ahead.
Relations between the two Koreas had chilled in recent weeks, with North Korea canceling a high-level meeting with Seoul over South Korea's participation in regular military exercises with the United States and insisting that it will not return to talks unless its grievances are resolved. Military talks and talks on meetings of divided families are also scheduled.
The agency said the two leaders had reached a "satisfactory consensus" and expressed "their stand to make joint efforts for the denuclearization of the Korean peninsula".