* First formal overture by Moon government

* Military talks would aim to halt hostile acts on border-S.Korea

* Talks proposed for July 21 at Panmunjom truce village

* Red Cross requests family reunion talks Aug 1 (Adds White Houses spokesman paragraphs 5-6)

By Christine Kim

SEOUL, July 17 (Reuters) - South Korea on Monday proposed military talks with North Korea, the first formal overture to Pyongyang by the government of President Moon Jae-in, to discuss ways to avoid hostile acts near the heavily militarised border.

There was no immediate response from North Korea to the proposal for talks later this week. The two sides technically remain at war but Moon, who came to power in May, has pledged to engage North Korea in dialogue as well as bring pressure to impede its nuclear and missile programmes.

The offer comes after North Korea claimed to have conducted the first test of an intercontinental ballistic missile (ICBM) earlier this month, and said it had mastered the technology to mount a nuclear warhead on the missile.

"Talks and cooperation between the two Koreas to ease tension and bring about peace on the Korean peninsula will be instrumental for pushing forth a mutual, virtuous cycle for inter-Korea relations and North Korea's nuclear problem," South Korea's Unification Minister Cho Myoung-gyon told a news briefing.

The United States, South Korea's main ally, which has been trying to rally international support for tougher sanctions on North Korea, appeared cool to the proposal, recalling President Donald Trump's statements that conditions must be right for dialogue.

"I think the President has made clear in the past with respect that any type of conditions that would have to be met are clearly far away from where we are now," White House spokesman Sean Spicer told a news briefing.

The South Korean defence ministry proposed talks with the North on July 21 at Tongilgak to stop all activities that fuel tension at the military demarcation line.

Tongilgak is a North Korean building at the Panmunjom truce village on the border used for previous inter-Korea talks. The last such talks were held in December 2015.

Cho also urged restoration of cross-border military and government hotlines that North Korea cut last year in response to South Korea's imposition of new economic sanctions after a nuclear test by Pyongyang.

South Korea also proposed separate talks by the rival states' Red Cross organisations to resume a humanitarian project to reunite families separated during the 1950-53 Korean War.

The South Korean Red Cross suggested talks be held on Aug. 1, with possible reunions over the Korean thanksgiving Chuseok holiday, which falls in October.