And Trump tweets that he's asked Moon about "Rocket Man" - an apparent reference to North Korean leader Kim Jong Un (kim jawng oon).
White House National Security Adviser H.R. McMaster said the United States was fast running out of patience for diplomatic solutions on North Korea's missile and nuclear programs. North Korea also recently tested its most powerful bomb.
Trump, who seeks to govern on an "America First" platform, has previously criticized the United Nations.
Later on Tuesday, Trump will have lunch with U.N. Secretary General Antonio Guterres, followed by meetings with Slovakia's Miroslav Lajčák, president of the U.N. General Assembly, and with the Emir of Qatar.
Asked on ABC News' "This Week" whether the country would remain signed on if better terms could be renegotiated before then, McMaster said: "If there's an agreement that benefits the American people, certainly".
Sen. Dianne Feinstein, who sits on the Senate Intelligence Committee, said she was very concerned about North Korea's rapid nuclear buildup.
McMaster said global support for diplomacy backed by rigorous enforcement of sanctions is needed to avoid what he called "the military option".
China has urged the United States to refrain from making threats to North Korea.
Haley told CNN's "State of the Union" program on Sunday that Washington had "pretty much exhausted" its options on North Korea at the Security Council.
Germany's UN Ambassador Christoph Heusgen said the most important thing about the General Assembly ministerial session, which officially begins Tuesday and ends September 25, is the opportunity for leaders to talk one-on-one or get together in groups.
On Friday, the U.N. Security Council condemned North Korea's "highly provocative" test of an intermediate-range missile, which hurtled over US ally Japan into the northern Pacific Ocean.
The top USA diplomat made a rare appearance on a Sunday talk show, where he said the US strategy is to pursue a "peaceful pressure campaign" based on what he called the four "nos": not seeking either regime change or collapse in North Korea, an accelerated re-unification of the Korean Peninsula, or a reason to send in military forces.
Harry Kazianis, director of defense studies at the Center for the National Interest in Washington, said Trump should use his General Assembly speech to increase pressure on China to do more to rein in North Korea's provocative actions.
She said that the USA wanted to be responsible and go through all diplomatic means to get the North's attention first, but if that doesn't work, Defense Secretary James Mattis will take care of it. "The latter two countries in particular have suffered direct and persistent attacks on their sovereignty in recent years", McMaster said Friday.
"I personally think he slaps the right people, he hugs the right people, and he comes out with (the) USA being very strong, in the end", Haley, speaking at the White House, said of Trump's speech.
BENNETT: But unlike past presidents, Trump is promoting his America-first agenda. Administration officials reaffirmed the United States' long-standing policy that the North must agree to relinquish its nuclear arsenal as a prerequisite for direct diplomatic talks.
And what have the US and its allies achieved?
Analysts say that in the event of hostilities, millions of people in the Seoul area - as well as the 30,000 United States troops in South Korea - would be vulnerable to attack by the thousands of artillery pieces the North has positioned near the border, with potentially staggering casualties.