Kim and Trump met three times since last year to negotiate an end to North Korea's nuclear and missile programmes but there has been scant progress. Working-level talks led by Biegun in October in Stockholm fell apart.
On Sunday, state media said North Korea had successfully conducted another test at a satellite launch site aimed at "restraining and overpowering the U.S. nuclear threat," a second such test in a week.
Seoul officials are concerned the flare-up may signal a repeat of earlier cycles of detente ending in escalations.
Biegun, who also met South Korean President Moon Jae-in on Monday, expressed regret over statements from North Korean officials in recent weeks, describing them as "so hostile and negative and so unnecessary".
"But it does not have to be this way. It is not yet too late," Biegun said.
Analysts said chances are slim for North Korea to accept Biegun's call for dialogue, citing Pyongyang's recent statements Washington had "nothing to offer" even if were talks to reopen.
"For North Koreans, Biegun effectively challenged their supreme leader by brushing off the year-end deadline," said Shin Beom-chul, a senior fellow at the Asan Institute for Policy Studies in Seoul.
Chun Yung-woo, a former South Korean nuclear envoy, said Biegun's offer appeared to be aimed at undercutting Pyongyang's bid to deal directly with Trump.
Trump's re-election battle and the impeachment inquiry against him may have led Kim to overestimate North Korea's leverage, diplomatic sources have told Reuters.
"North Korea won't likely show up at the negotiating table unless the United States promises to make clear concessions instead of calling for a meeting for meeting's sake," said Kim Dong-yub, a professor at Kyungnam University in Seoul.
