The planned inter-Korean summit follows Trump's cancellation of a visit to Pyongyang by U.S. Secretary of State Mike Pompeo last month after Pompeo received a belligerent letter from a senior North Korean official. The U.N. General Assembly in late September would be an ideal date for Seoul, but many analysts see that possibility as low, considering the complications of the process and how far apart the parties now are.
North Korea wants the United States to formally declare an end to the Korean War before it takes any concrete steps to open up or scale back its nuclear program, arguing that such a step is vital to defusing tension on the Korean Peninsula and building trust.
"Our government believes that an end-of-war declaration is very much needed while we enter a process toward stabilizing peace in the Korean Peninsula through complete denuclearization", said Chung Eui-yong, Moon's national security adviser and the head of the South Korean delegation to Pyongyang, in a news conference on Tuesday.
Moon and Trump agreed in their call "to explore the idea of meeting in person on the margins of the U.N. General Assembly and having in-depth consultations on strategies and how to cooperate on the peninsula issues", Moon's office said in a statement. The North, meanwhile, could easily find excuses to drag out the process or even discard it.
Moon is also running out of goodwill gestures toward North Korea, which has expressed frustration about the slow pace in cooperative projects with the South that it hopes will bring it economic benefits.
Moon's approval rating, although still above 50 percent, has declined sharply in recent months over a decaying job market.
At the summit the pair reached a vague agreement to work towards the "complete denuclearisation of the Korean peninsula", but there has been little movement since.
"If inter-Korean relations also stagnate, the diplomatic momentum could be lost", he said in an interview.
The Pyongyang summit could reveal whether Moon is ready to push ahead with more engagement with the North despite US frustration over North Korea.
Impoverished and isolated North Korea has prioritised its nuclear weapons, achieving remarkable success in recent years.