By Jack Kim and Ju-min Park

SEOUL (Reuters) - U.S. President-elect Donald Trump pledged his commitment to defending South Korea under an existing security alliance during a phone call with South Korean President Park Geun-hye on Thursday, her office said.

Trump had said during the election campaign he would be willing to withdraw U.S. military stationed in South Korea unless Seoul paid a greater share of the cost of the U.S. deployment, but an adviser to the president-elect played down such comments on Thursday.

There are about 28,500 U.S. troops based in South Korea helping to defend the country against nuclear-armed North Korea, which has remained in a technical state of war with the South since the 1950-53 Korean conflict.

Park said the U.S.-South Korean alliance had grown in the past six decades and she hoped it would develop further.

She asked Trump to join in the effort to help minimize the threat from North Korea, which has carried out repeated nuclear and missile tests in defiance of U.N. Security Council resolutions and sanctions.

South Korea's presidential Blue House said Trump agreed with Park and it quoted Trump as saying: "We will be steadfast and strong with respect to working with you to protect against the instability in North Korea."

Trump's transition team did not respond to requests for comment, but an adviser to the president-elect played down concerns about any changes in approach to alliances in Asia.

"I think what we are going to see is a very mainstream Republican administration," he said, adding that sharing the costs of sustaining combined defense was a matter for negotiation, as it always had been.

"It’s going to be a respectful conversation that’s going to be done at the working level and will have absolutely nothing to do with the overall strength of these alliances, which is going to remain extremely close," he said.

The adviser, who did not want to be identified by name, also noted that during the election campaign Trump had dropped comments he had made saying he would consider letting Japan and South Korea build their own nuclear weapons rather than have them rely on the United States nuclear umbrella.

"He has moved on to talk about non-proliferation in a way that you would hear from any Republican president," he said.

"We are very much committed to both non-proliferation and assuring the allies that not only will they continue to be under the nuclear umbrella, but that we are going to be strengthening our missile defense in ways that alleviate some of their concerns about North Korea."

The Blue House said the call between Park and Trump lasted about 10 minutes and Park said she hoped Trump would be able to visit South Korea soon.

COST SHARING CONCERNS

There has been concern in South Korea that a Trump presidency will demand that Seoul sharply raise its share of the cost of maintaining the U.S. military presence.

Under a five-year cost-sharing accord reached two years ago, Seoul agreed to contribute $867 million toward U.S. military costs in 2014, about 40 percent of the total. The deal called for the amount to rise annually at the rate of inflation.

Trump said earlier this year that the United States was paid "peanuts" for the troop presence and that he would be willing to withdraw U.S. forces from South Korea and Japan, but "would not do so happily."

South Korea believes its share of the cost is much higher when the vast amount of land occupied by the U.S. forces, including a large area in central Seoul, are considered.

Some members of the South Korean parliament have suggested that the country has little choice but to consider nuclear armament if U.S. forces are withdrawn while North Korea continues to develop nuclear weapons and missiles that could carry them.

South Korea's Defense Ministry spokesman Moon Sang-gyun said on Thursday the country paid its share of the cost of maintaining the U.S. military and the contribution had been recognized by the U.S. government and Congress.

South Korea and the United States have also agreed to deploy a Terminal High Altitude Area Defense (THAAD) anti-missile system with the U.S. military to counter missile threats from North Korea.

South Korea has consistently said it had no plan to buy the THAAD system, which is built by Lockheed Martin Corp and costs an estimated $800 million a piece, that will likely add to the cost of maintaining the U.S. military presence.

The official newspaper of the North's ruling Workers' Party said on Thursday the U.S. wish for North Korea to abandon its nuclear weapons program was "only a fantasy of a bygone era" and the policy of pressure and sanctions had failed.

"The only accomplishment of the Obama administration is that it is leaving behind for the new administration coming next year the burden of having to deal with a strong nuclear power," Rodong Sinmun said in a commentary.

It did not mention Trump by name. But Choson Sinbo, a pro-North Korean newspaper published in Japan and controlled by Pyongyang, said: "Trump is well advised to learn the lesson of history from Obama's failure.

"Otherwise, the new owner of the White House will be met with the ashes of the calamity started by the previous owner."

(Additional reporting by David Brunnstrom in Washington; Editing by Nick Macfie and Bill Trott)