United States President Joe Biden opened his virtual meeting with Chinese President Xi Jinping on Monday by saying the goal of the two world leaders should be to ensure that competition between the two superpowers "does not veer into conflict."

Xi greeted the U.S. president as his "old friend" and echoed Biden’s cordial tone in his own opening remarks, saying, "China and the United States need to increase communication and cooperation."

While the two leaders opened their meeting with friendly waves, they were meeting at a time of mounting tensions in the U.S.-China relationship. Biden has criticized Beijing for human rights abuses against Muslim Uyghurs in northwest China, suppression of democratic protests in Hong Kong, military aggression against the self-ruled island of Taiwan and more. Xi’s deputies, meanwhile, have lashed out against the Biden White House for interfering in what they see as internal Chinese matters.

"It seems to be our responsibility as the leaders of China and the United States to ensure that the competition between our countries does not veer into conflict, whether intended or unintended, rather than simple, straightforward competition," Biden said at the start of the meeting.

The two leaders know each other well, having traveled together when both were vice presidents.

"I stand ready to work with you, Mr. President, to build consensus, take active steps and move China-U.S. relations forward in a positive direction," Xi said, according to The Associated Press (AP).

The talks, which were initiated by Biden and began at 7:46 p.m. on Monday (12:46 a.m. GMT Tuesday), were intended to make the relationship less acrimonious. The two sides took a 15-minute break after a nearly two-hour first session that ran half an hour longer than expected, according to reports from Chinese state media, before resuming the conversation. The U.S. president was joined in the Roosevelt Room for the video call by Secretary of State Antony Blinken and a handful of aides. Xi, for his part, was accompanied by communist party director Ding Xuexiang and a number of advisers.

The high-level diplomacy had a touch of pandemic Zoom meeting informality as the two leaders waved to each other once they saw one another on the screen, with Xi telling Biden, "It’s the first time for us to meet virtually, although it’s not as good as a face-to-face meeting."

Biden would have preferred to meet Xi in person, but the Chinese leader has not left his country since the start of the coronavirus pandemic. The White House floated the idea of a virtual meeting as the next best thing to allow for the two leaders to have a candid conversation about a wide range of strains in the relationship.

Chinese officials said in advance that Taiwan would be their top issue for the talks. Tensions have heightened as the Chinese military has dispatched an increasing number of fighter jets near the self-ruled island of Taiwan, which Beijing considers part of its territory.

"The Taiwan issue concerns China’s sovereignty and territorial integrity, as well as China’s core interest," Chinese Foreign Ministry spokesperson Zhao Lijian said Monday. "It is the most important and sensitive issue in China-U.S. relations."

The White House said Biden will abide by the longstanding U.S. "One China" policy, which recognizes Beijing but allows informal relations and defense ties with Taipei. Chinese military forces held exercises last week near Taiwan in response to a visit by a U.S. congressional delegation to the island.

According to Agence France-Presse (AFP) Xi Jinping warned Biden that U.S. support for Taiwanese independence would be "like playing with fire," state media said Tuesday. The two also spoke about key regional challenges, including North Korea, Afghanistan and Iran.

"Taiwanese authorities have repeatedly tried to 'rely on the U.S. for independence,'" Xi was quoted as saying by state media agency Xinhua, adding: "Some people in the U.S. intend to 'use Taiwan to control China.'

"This trend is very dangerous and is like playing with fire, and those who play with fire will get burned."

The White House readout after the summit was considerably more measured, but between the lines, Biden's pushback against Beijing's increasingly aggressive posture toward Taiwan was clear.

"On Taiwan, President Biden underscored that the United States ... strongly opposes unilateral efforts to change the status quo or undermine peace and stability across the Taiwan Strait," the White House statement said.

With Beijing set to host the Winter Olympics in February and Xi expected to be approved by Communist Party leaders to serve a third five-year term as president next year – unprecedented in recent Chinese history – the Chinese leader may be looking to stabilize the relationship in the near term. White House press secretary Jen Psaki added that the "condensing of power" in China made the leader-to-leader conversations essential. Slowing economic growth and a brewing housing crisis also loom large for Beijing.

"China and the United States should respect each other, coexist in peace, and pursue ... cooperation," Xi said.

At the same time, Biden, who has seen his polling numbers diminish at home amid concerns about the lingering coronavirus pandemic, inflation and supply chain problems, was looking to find a measure of equilibrium on the most consequential foreign policy matter he faces. Despite his domestic problems, White House officials made the case that Biden was coming into his meeting with Xi from a position of strength.

Earlier Monday, Biden signed into law a $1 trillion infrastructure bill, legislation to upgrade the nation’s crumbling infrastructure that the Democrat argued is crucial for the U.S. as it seeks to retain a competitive edge over China.

"Because of this law, next year will be the first year in 20 years American infrastructure investment will grow faster than China’s," Biden said earlier Monday. "We will once again have the best roads, bridges, ports and airports over the next decade."

The White House set low expectations for the meeting with Xi: No major announcements or even a joint statement were anticipated. The U.S. president has held up his relationship with Xi as evidence of his heartfelt belief that good foreign policy starts with building strong personal relationships.

The public warmth – Xi referred to Biden as his "old friend" when the then-vice president visited China in 2013, while Biden spoke of their "friendship" – has cooled now that both men are heads of state. Biden bristled in June when asked by a reporter if he would press his old friend to cooperate with a World Health Organization (WHO) investigation into the coronavirus origins.

"Let’s get something straight: We know each other well; we’re not old friends," Biden said. "It’s just pure business."

Xi, however, at the start of the meeting gave a nod to the warmth in the earlier days of their relationship, saying, "I am very happy to see my old friend."

Biden said he expected the two would spend plenty of time discussing areas of difference, including human rights, economics and "ensuring a free and open Indo-Pacific."

"We have always communicated with one another very honestly and candidly. We never walk away wondering what the other man is thinking," Biden said.

The first nine months of the Biden administration have been marked by the two sides trading recriminations and by unproductive exchanges between the presidents’ top advisers. But there are signs of thawing.

Last week, the U.S. and China pledged at U.N. climate talks in Glasgow, Scotland, to increase their cooperation and speed up action to rein in climate-damaging emissions.

Republicans have accused the Biden administration of failing to hold Beijing accountable on human rights for the sake of pursuing its climate agenda.

"As he turns a blind eye to human rights atrocities to pursue his political agenda, Biden has allowed China to threaten American security and our allies’ sovereignty while undermining the advancement of freedom across the globe," the Republican National Committee said in a statement shortly before the start of the leaders’ meeting.

The White House has said it views cooperation on climate change as something in China’s interest, something the two nations should cooperate on despite differences on other aspects of the relationship.

"None of this is a favor to either of our countries, what we do for one another, but it is ... responsible world leadership," Biden told Xi. "You’re a major world leader, so is the United States."