UPDATE 10-Trump offers North Korea's Kim weekend meeting in demilitarized zone

* Trump makes offer of brief handshake in tweet

* N.Korea official: meeting could advance two-way ties -KCNA

* S.Korean president Moon welcomes Trump's initiative

* Trump and Kim have exchanged letters (Adds Moon comment, paragraph 14, edits)

By Roberta Rampton and Joyce Lee

OSAKA/SEOUL, June 29 (Reuters) - U.S. President Donald Trump on Saturday proposed a weekend meeting with Kim Jong Un at the demilitarized zone (DMZ) between North and South Korea, an encounter North Korea said would be meaningful if it goes ahead.

If Trump and Kim do meet, it will be for the third time in just over a year, and four months after their second summit broke down with no progress on U.S. efforts to get North Korea to give up its nuclear weapons.

North Korea said it had not had an official proposal.

Trump made the offer in a tweet about his trip to South Korea, where he landed on Saturday after the Group of 20 summit in Osaka, Japan.

"While there, if Chairman Kim of North Korea sees this, I would meet him at the Border/DMZ just to shake his hand and say Hello(?)!" Trump said, later telling reporters his offer was spontaneous: "I just thought of it this morning."

"If he's there, we'll see each other for two minutes, that's all we can, but that will be fine," he said, adding that he and Kim "get along very well".

About five hours after Trump's tweet, the state news agency KCNA quoted Choe Son Hui, North Korea's first vice-minister of foreign affairs, as saying: "We see it as a very interesting suggestion, but we have not received an official proposal."

"If the DPRK-U.S. summit meetings take place on the division line, as is intended by President Trump, it would serve as another meaningful occasion in further deepening the personal relations between the two leaders and advancing the bilateral relations," Choe said in a statement.

She was referring to North Korea by its official name - the Democratic People's Republic of Korea.

"KIM IS RECEPTIVE"

Later, Trump told a news conference: "We may be meeting with Chairman Kim ... Kim Jong Un was very receptive."

He added, "We won't call it a summit. We'll call it a handshake", and said he would be very comfortable stepping over the border into North Korea if he met Kim at the DMZ.

After arriving in Seoul, Trump was asked whether there would be a three-way meeting with Kim and South Korean President Moon Jae-in on Sunday. "We're working things out right now," he said. "We’re going to see."