The two leaders discussed the issue as a way to boost the South's defence against growing threat from North in a telephone call, South Korea's Blue House said.
South Korea and the U.S. are discussing the dispatch of F-22 and F-35B fighter jets on a rotational basis as part of efforts to boost extended deterrence against the North's threats, they said.
Trump, who has warned that the US military is "locked and loaded" in case of further North Korean provocation, reacted angrily to the latest missile test, declaring on Twitter that "talking is not the answer" to resolving the crisis.
"President Moon Jae-In ordered the toughest retaliation measures against North Korea's ICBM level missile and nuclear provocations, mobilizing all diplomatic means to totally isolate North Korea", Chung Eui-Yong, director of the national security office in the South Korean president's office, said on Sunday in a nationally televised press briefing. The two agreed that the USA and South Korean presidents would talk on the phone soon to discuss cooperation on measures against North Korean provocation, he said.More news: Nurse Refuses Blood Draw On Unconscious Patient "You're Under Arrest'"
"The two leaders agreed to strengthen our alliance through defense cooperation and to strengthen South Korea's defense capabilities".
Trump also gave "conceptual" approval to the purchase by the South of billions of dollars of United States military hardware, the White House said.
The South wants the maximum warhead weight doubled to one tonne, and the Pentagon has said it was "actively" considering the revision. Pacific Command and South Korea's defense minister, a Pacom spokeswoman said.
The quake detected around the North's Punggye-ri nuclear test site came after Seoul's intelligence officials said that Pyongyang appears ready for what would be its sixth nuclear test following the two tests a year ago.More news: United States forces Russian Federation to close San Francisco consulate, shrink 2 others
On Friday, South Korean President Moon Jae-in reportedly held a telephonic conversation with U.S. President Donald Trump.
This photo, provided by the presidential office on August 21, 2017, shows President Moon Jae-in speaking during a National Security Council session at the presidential office Cheong Wa Dae in Seoul.
The South, which hosts 28,500 United States troops to defend it, is banned from building its own nuclear weapons under a 1974 atomic energy deal it signed with Washington, which instead offers a "nuclear umbrella" against potential attacks.
However, Trump said after the latest missile test that negotiations with Pyongyang were "not the answer".More news: Taylor Swift's new music video makes biggest YouTube debut ever