The draft resolution also called for a ban to be lifted on North Koreans working abroad and the termination of a 2017 requirement for all such workers to be repatriated by next week.
The two leaders agreed that the situation has become "severe" and that "dialog momentum should be maintained to achieve prompt results from denuclearisation negotiations."
Kim Jong Un has set an end-of-the-year deadline for denuclearisation talks with Washington.
In a speech at the U.N. General Assembly, Pyongyang's U.N. ambassador Kim Song said it was time for Washington to share proposals for talks.
The launches followed repeated short-range missile tests by North Korea since its leader Kim Jong Un agreed in a June 30 meeting with U.S. President Donald Trump to reopen working-level talks
Saturday's launch was the seventh by North Korea since U.S. President Donald Trump and North Korean leader Kim Jong Un met at the inter-Korean border in June.
"Kim Jong Un has been very straight with me," Trump told reporters.
North Korea remains unchanged in its position to resolve all issues through dialog and negotiation, a North Korean Foreign Ministry spokesman said.
The North has protested against joint US-South Korea military drills, which kicked off last week, calling them a rehearsal for war.
North Korea fired missiles into the sea off its east coast for the fourth time in less than two weeks, the South Korean military said on Tuesday.