(CNN) President Donald Trump, arriving at the United Nations on Monday morning, said he and Kim Jong Un will meet for a second time "quite soon."

"It looks like we'll have a second summit quite soon. As you know, Kim Jong Un wrote a letter -- a beautiful letter -- and asked me for a second meeting and we will be doing that. (Secretary of State Mike) Pompeo will work that out in the immediate future," Trump said, claiming there has been "tremendous progress on North Korea."

Diplomacy between the US and North Korea has continued in fits and starts since Trump met with Kim in Singapore in June, with Trump most recently canceling Pompeo's planned trip to Pyongyang last month, believing the visit would not be fruitful.

Trump's announcement follows weeks of encouragement on the part of North and South Korea for the US President to continue his diplomatic outreach to Pyongyang.

Trump has continued to express a rosy optimism about the fate of the US-North Korea diplomatic effort and repeatedly expressed confidence in the strength of his personal relationship with Kim. The second summit, Trump said later in the day, would take place at an undetermined location, though likely not in Singapore, where the leaders first met in June.

Read More