With a few apparently off-the-cuff comments, US President-elect Donald Trump has threatened Washington's cautious understanding with China while touting an unlikely new detente with Russia.

The United States and China, the world's two greatest economies and rivals for the leadership role in the Pacific, are often at loggerheads over trade, human rights and regional disputes.

But President Barack Obama has extended a hand to China's Xi Jinping and worked with Beijing on the global climate change accord and on measures against North Korea's rogue regime.

On Monday, businesswoman turned defeated presidential hopeful Carly Fiorina left Trump's New York office and said they had discussed Trump's opportunity to "reset" US foreign relations.

As part of this, Fiorina told reporters, she and Trump "spent a fair amount of time talking about China as probably our most important adversary and a rising adversary."

Never has current US leader Obama called into question the "One China" doctrine, which accepts that currently self-administered Taiwan is part of one state one day to be united under Beijing.

Trump may have recently taken advice from Henry Kissinger, architect of late president Richard Nixon's US-China breakthrough, but he has broken with this four-decade consensus.

Over the weekend, in a series of Tweets and an interview with Fox News, the incoming president suggested Beijing may have to make concessions on trade if US policy is not to change.

The tactic drew scorn from the outgoing US administration. Obama's spokesman Josh Earnest said Taiwan's future should not become a "bargaining chip" in US relations with Beijing.

In directly challenging China over trade and its takeover of disputed islets in the South China Sea, Trump risks provoking Beijing into a response with global economic implications.

"I don't know why we have to be bound by a One China policy unless we make a deal with China having to do with other things, including trade," Trump told Fox News.

In this, he was doubling down on an earlier provocation when he took a call from Tsai Ing-wen, the elected president of Taiwan, in a break with established US protocol of non-recognition.

The response from China -- America's biggest single goods trading partner and the holder of almost a trillion dollars in US government debt -- was ominous.

Chinese foreign ministry spokesman Geng Shuang said Beijing considers the island a "core interest" in China's sovereignty and territorial integrity.

The One China policy is the "political bedrock" of relations with the US, he added and if it is "compromised or disrupted" cooperation in major fields would be "out of the question."

- 'Childish ignorance' -

This firm warning was echoed in Chinese state media, including in an editorial in the Global Times that dubbed the famously thin-skinned Trump "as ignorant of diplomacy as a child."

Trump has also said he will declare China, which he accuses of keeping the Renminbi artificially low to favor its exports, a currency manipulator, forcing a renegotiation of trade ties.

If he makes good on his campaign promises to tax US companies that move plants offshore and to impose tariffs on cheap imported goods, Trump could provoke a trade war.

In the meantime, Trump has pledged to walk away from the Trans-Pacific Partnership (TPP) trade deal, disappointing allies like Japan and ceding ground to China in the Pacific leadership stakes.

When Kissinger and Nixon launched their detente with China 45 years ago it was in part to ensure Washington was not fighting the Cold War on two fronts against both Beijing and Moscow.

Now, Trump appears to be trying to flip this paradigm on its head by needling China while showing all signs of wanting to thaw relations with Vladimir Putin's aggressive Russia.

He has pushed back furiously at the suggestion in leaks from the CIA that Putin's intelligence services hacked US political parties and planted false news to further Trump's campaign.

After her chat with Trump, Fiorina even tried to flip the focus of the scandal by saying they discussed "hacking, whether it's Chinese hacking or purported Russian hacking."

- Order of friendship -

Trump has defended his desire to have better relations with Russia, arguing it would be a good partner in the fight against the Islamic State group, despite Putin's annexation of Crimea.

Last week, Putin reciprocated Trump's evident enthusiasm for closer ties, dubbing him a "clever person" who will "fully and quite quickly grasp a different level of responsibility."

And, while clues to the detail of Trump's future foreign policy are lacking, his choice of personnel speaks volumes.

Trump's national security adviser will be former US defense intelligence chief Mike Flynn, reportedly forced into retirement by Obama's administration for his erratic style.

Flynn was a guest at a 2015 Moscow dinner held by Russia's state television network RT to honor Putin, and has defended Moscow from criticism of its campaign in Syria.

Meanwhile, ExxonMobil CEO Rex Tillerson, favorite to be named Trump's secretary of state, opposed sanctions on Russia and in 2013 was awarded the Order of Friendship by Putin.