SEOUL (Reuters) - South Korea proposed talks with North Korea on how to handle the North's Mt Kumgang resort, once a rare example of cooperation between the Koreas, which Pyongyang wants removed amid frayed ties, Seoul's Unification Ministry said on Monday.

North Korean leader Kim Jong Un said last week he wanted the "shabby" and "capitalist" facilities removed and rebuilt in the latest sign of cooling relations between the two Koreas.

The Unification Ministry, which oversees inter-Korean affairs, said it asked the North to hold a working-level meeting to discuss the issue. Pyongyang suggested last week the subject be handled through the exchange of documents.

"Our consistent position is that all issues of inter-Korean relations should be resolved through dialogue and discussions," ministry spokesman Lee Sang-min told a briefing.

"There needs to be sufficient consultations to reasonably resolve the issue since a unilateral step regarding our companies' property rights runs counter to public sentiment and may damage inter-Korean relations," he said.

The proposal did not specify a date or venue for talks, Lee said. It was delivered to two state agencies, each handling cross-border issues and tourism, through a liaison office in the North's border city of Kaesong.

Any pullout of South Korean facilities from the scenic resort would be a fresh blow for President Moon Jae-in's attempts to promote peace between the old foes, including efforts to kick-start stalled business projects.

Mt Kumgang, located on the North's eastern shore just beyond the heavily fortified border separating the two Koreas, is one of two major inter-Korean economic initiatives, along with the Kaesong factory park.

Inter-Korean ties have grown strained this year, with the North shunning formal talks with the South, amid stalled progress in denuclearization talks between Pyongyang and Washington.

Lee said the government could not predict whether the North would respond to its offer.

A coalition of some 200 civic groups working on inter-Korean peace, better known as Minhwahyup, issued a statement on Monday calling for the Kumgang and Kaesong programs to be restarted.

(Reporting by Hyonhee Shin; Editing by Paul Tait)