WASHINGTON, May 21 (Reuters) - U.S. President Joe Biden and South Korean President Moon Jae-in on Friday injected fresh urgency into attempts to engage North Korea in dialogue over its nuclear weapons, with Biden saying he would meet its leader Kim Jong Un under the right conditions.

At a joint news conference, Biden and Moon both said the complete denuclearization of the Korean peninsula is their goal. Biden said he was "under no illusions" about the difficulty of getting North Korea to give up its nuclear arsenals after his predecessors failed.

"We both are deeply concerned about the situation," Biden said, adding that he and Moon shared a willingness to engage diplomatically with the North "to take pragmatic steps to reduce tensions."

North Korea thus far has rebuffed U.S. entreaties for diplomacy since Biden took over from Donald Trump, who had three summits with Kim and the two famously exchanged "beautiful letters."

Kim nonetheless refused to give up his nuclear weapons but did impose a freeze on testing them. He has not tested a nuclear bomb nor launched an inter-continental ballistic missile since 2017, although experts believe his arsenal has steadily grown.

Biden said he would be willing to meet Kim under the right conditions - if he agreed to discuss his nuclear program and that his advisers first met with their North Korean counterparts to lay the groundwork.

Biden said a veteran State Department official, Sung Kim, would serve as special U.S. envoy for North Korea.

South Korea had been pushing for the appointment of an envoy, and Moon, for whom engagement with North Korea is a legacy issue before he leaves office next year, said Sung Kim would help explore whether North Korea is willing to engage diplomatically. He said he expected a positive response.

Sung Kim is a Korean-American diplomat who served as special envoy for North Korea under President Barack Obama and helped set up Trump's summits with Kim Jong Un.

He has also been ambassador to South Korea, the Philippines and Indonesia and most recently served in an acting capacity as the top U.S. diplomat for East Asia.

Biden said that for him to meet with Kim there would have to be a commitment from the North Korean leader "that there's discussion about his nuclear arsenal."

Biden's comments appeared to reflect a shift in his thinking. The White House had said in March it was not Biden's intention to meet with Kim.

Moon said Biden had expressed support for intra-Korean dialogue and cooperation, but Town said this would not mean much unless Washington, which has refused to ease sanctions on Pyongyang, allowed Seoul the space to move forward with this.

Biden's administration undertook a broad review of North Korea policy but has said little about what its new policy actually entails.

U.S. officials have said only that it would not be the approach favored by Obama of refusing to engage the North, or Trump's flashy summitry.

In their day of talks, Biden and Moon reaffirmed a strong alliance between the two countries after the strains created by Trump, who badgered Moon as weak and threatened to pull U.S. troops out of South Korea.

Moon was the second foreign leader - after Japan's prime minister - to visit the White House since Biden took office in January, and Biden said their conversations were those of "old friends."

The two also discussed China and Taiwan. Taiwan has complained of repeated military pressure from Beijing with China’s air force making frequent forays into Taiwan’s air defense identification zone.

In another win for Moon, who has faced pressure at home over his handling of the COVID-19 pandemic, he said the United States and South Korea had agreed a vaccine partnership that would combine U.S. expertize and Korean production capacity.

He said this would help supply COVID-19 vaccines in the Indo-Pacific region, while Biden said the United States would provide vaccinations for 550,000 South Korean soldiers.