(Adds Tillerson comments)
* China says U.S. comments on North Korea "positive"
* Putin, Abe call on Pyongyang and others to avoid provoking tensions
* North Korea official says won't stop nuclear tests in face of U.S. "hostile acts"
By Michael Martina and David Brunnstrom
BEIJING/WASHINGTON, April 27 (Reuters) - China has told the Trump administration it warned North Korea it would impose unilateral sanctions should Pyongyang carry out another nuclear test, U.S. Secretary of State Rex Tillerson said on Thursday, in possibly China's toughest threat yet against its defiant neighbour.
Tillerson, speaking on Fox News, said China has asked North Korea not to conduct any more nuclear tests. He noted that major commemorations in North had passed in recent days without a feared nuclear test or test of any intercontinental ballistic missile.
"And in fact we were told by the Chinese that they informed the regime that if they did conduct further nuclear tests, China would be taking sanctions actions on their own," Tillerson said, without specifying what sanctions he was referring to.
Tillerson did not say when China made the threat and there was no immediate confirmation from Beijing. He is due to chair a meeting with U.N Security Council foreign ministers on Friday, where he said he would stress the need for members to fully implement existing sanctions as well as possible next steps.
"We're going to be discussing what next steps may be necessary to increase the pressure on the regime," he said.
The Chinese embassy in Washington did not immediately respond to a request for comment.
Tillerson's comments suggest that President Donald Trump's administration, which held a summit with Chinese President Xi Jinping this month, believes its efforts may have yielded a stronger line by Beijing against the North.
China banned imports of North Korean coal in February, cutting off its most important export, and Chinese media this month raised the possibility of restricting oil shipments to the North if it unleashed more provocations.
China earlier on Thursday welcomed an apparently softer tone by the United States on the North Korean nuclear and missile crisis but stressed its opposition to a U.S. missile defence system being deployed in South Korea.
China has long promoted dialogue to resolve the Korean nuclear issue as North Korea has repeatedly threatened to destroy the United States. Washington has in turn warned that "all options are on the table" in ending North Korean provocations.
But Tillerson was quoted telling National Public Radio as saying the United States would favour direct talks with the North, if it has "the right agenda."
On Fox News, Tillerson said U.S. intelligence reports suggest North Korean leader Kim Jong Un was "not crazy," indicating he might be a rational actor who could be negotiated with.
"He may be ruthless, he may be a murderer. He may be someone who in many respects we would say by our standards is irrational. But he is not insane," Tillerson said.
North Korea's growing nuclear and missile threat is a major security challenge confronting Trump, who has vowed to prevent North Korea from being able to hit the United States with a nuclear missile, a capability experts say Pyongyang could have some time after 2020.
Trump's administration said on Wednesday it aimed to push North Korea into dismantling its nuclear and missile programmes, which are in violation of U.N. Security Council resolutions, through tougher international sanctions and diplomatic pressure.
Asked about the U.S. comments, Chinese Foreign Ministry spokesman Geng Shuang said China had noted that many U.S. officials had recently made such remarks.
"We have noted these expressions, and have noted the message conveyed in these expressions hoping to resolve the Korean nuclear issue peacefully through dialogue and consultation," he said. "We believe this message is positive and should be affirmed."
Separately, Russian President Vladimir Putin and Japanese Prime Minister Shinzo Abe called on North Korea and other countries on Thursday to avoid behaviour or rhetoric that could increase tensions around Pyongyang's nuclear programme.
South Korea and the United States agreed on Thursday on "swift punitive measures" against North Korea in the event of further provocation. The South also said the deployment of a U.S. anti-missile defence system was moving ahead effectively a day after angry protests by local residents against the battery and amid fierce opposition from China.
South Korea on Wednesday moved parts of the Terminal High Altitude Area Defense (THAAD) system to its deployment site on a golf course about 250 km (155 miles) south of the capital, Seoul, signalling faster installation.
Several hundred villagers protested, hurling water bottles at vehicles moving the parts in.
The top U.S. Commander in the Pacific, Admiral Harry Harris, said on Wednesday the THAAD system would be operational "in coming days," bolstering the ability to defend the U.S. ally and the 28,500 U.S. troops stationed there.
A photograph of the site showed a THAAD interceptor on a mobile launcher erected and pointed skywards on a green lawn as a military transport helicopter hovered nearby.
China says the system's advanced radar can penetrate deep into its territory and undermine its security. It is adamant in its opposition.
"The deployment of the THAAD anti-missile system in South Korea damages the regional strategic balance and stability. The Chinese side is resolutely opposed to this," Defence Ministry spokesman Yang Yujun told reporters.
"China’s military will continue to carry out live-fire military exercises and test new military equipment in order to firmly safeguard national security and regional peace and stability."
Susan Thornton, acting assistant secretary of state for East Asia, said on Thursday the U.S. policy was to exert maximum pressure on North Korea through sanctions and diplomatic activity "as frankly a last best way of seeing whether we can get a peaceful resolution to this problem."
Additional reporting by Phil Stewart, David Brunnstrom, Matt Spetalnick, Eric Beech and Patricia Zengerle in Washington and Ben Blanchard in Beijing; Writing by Jack Kim, Nick Macfie, Frances Kerry and Phil Stewart; Editing by Robert Birsel and James Dalgleish