South Korea has offered to hold talks with North Korea to ease animosities along their tense border and resume reunions of families separated by their war in the 1950s.

"We request military talks with the North on July 21 at Tongilgak to stop all hostile activities that raise military tension at the military demarcation line," Suh Choo-suk, South Korea's vice defence minister, told a media briefing on Monday.

South Korea's proposal for two sets of talks indicates new President Moon Jae-in is pushing to improve ties with North Korea, despite the North's first intercontinental ballistic missile test earlier this month.

READ MORE: Who is Moon Jae-in, South Korea's new president?

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

The proposal came roughly a week after Moon had said the need for dialogue with North Korea was more pressing than ever to curb North Korea's nuclear and missile programmes.

The vice defence minister did not elaborate on the meaning of hostile military activities, which are viewed differently by the two Koreas. South Korea usually refers to loudspeaker broadcasts and other provocations, while North Korea wants a halt to routine joint US-South Korea military drills.

Moon has suggested hostile military activities be halted at the inter-Korean border on July 27, the anniversary of the 1953 armistice agreement that ended the Korean War.

Talks on family reunions

South Korea's Red Cross says it wants separate talks at the border village on August 1 to discuss family reunions, with possible reunions over the Chuseok holiday, which falls in October this year.

North Korea's state media has not immediately responded to South Korea's latest overtures.

Previously, North Korea has repeatedly said it refuses to engage in talks unless South Korea turns over 12 waitresses who defected to the South last year.

North Korea says South Korea abducted the 12 waitresses and the restaurant manager and has demanded their return, but South Korea has said the group decided to defect of its own free will.

Al Jazeera's Kathy Novak, reporting from Seoul, said family separation is seen as a "very pressing and emotional humanitarian issue," - particularly in South Korea.

"We know that they are very sensitive about this issue because these people are getting older - effectively time is running out for them to have the opportunity to meet their family members again," she said.

"And the other thing that the unification minister has been talking about is a proposal to allow these people to visit ancestral graves for the first time."

Our correspondent said the military talks would be held first.

"And perhaps, if that went well, then it would set the scene for the Red Cross to be able to talk a bit more about the possibility of having family reunions this year," she said.

READ MORE: Three things to know about North Korea's missile tests

North Korea has conducted two nuclear tests since the beginning of last year and missile-related activities at an unprecedented pace.

It conducted the first test of an intercontinental ballistic missile (ICBM) earlier this month, claiming to have mastered the technology to mount a nuclear warhead on the missile. South Korea and the United States dispute the claim.

In an bid to rein in North Korea, the US is preparing new sanctions on Chinese banks and firms doing business with North Korea, possibly within weeks, two senior US officials said last week.