The leaders of the world's two largest economies met on the sidelines of an Asia-Pacific summit in Lima overshadowed by concerns of new trade wars and diplomatic tensions when Trump returns to the White House in January.
After shaking hands to begin their meeting at the Chinese leader's hotel in the Peruvian capital, Xi noted that Beijing's goals for stable ties with Washington remained unchanged.
"China is ready to work with the new US administration to maintain communication, expand cooperation and manage differences, so as to strive for a smooth transition of the China-US relationship," he said in translated comments.
Xi and Biden have led efforts to ease tensions over issues from trade to Taiwan, and Saturday's face-to-face was their second meeting since a landmark ice-breaking summit in California last year -- their third overall.
Biden said he was "very proud of the progress we both made together."
"Our two countries cannot let any of this competition veer into conflict. That's our responsibility and over the last four years I think we've proven it's possible to have this relationship," Biden said.
- 'Turbulent world' -
Trump's impending return to the Oval Office, with several China hawks in his top team, has stirred fears that tensions could return.
In his first White House term, Trump engaged in a bruising trade war with China, imposing tariffs on imports that saw retaliation from Beijing.
Trump has once again signaled a confrontational approach, threatening during the campaign against Democrat Kamala Harris to impose tariffs of up to 60 percent on imports of Chinese goods.
But in pointed comments, Xi said that China and the United States should "inject more certainty and positive energy into the turbulent world."
The White House said Biden had raised several key points of tension, including self-governed Taiwan -- which China claims as part of its territory and has repeatedly threatened.
Biden expressed "deep concern" over Beijing's support of Russia's defense industry as Moscow invades Ukraine, and condemned the deployment of thousands of troops to Russia by China's ally North Korea, the White House said in a readout of the meeting.
But it was the transfer of power in Washington that hung heaviest over the meeting, with Biden and Xi both seeking to calm potentially choppy waters.
US National Security Advisor Jake Sullivan told reporters Biden had reinforced that it was a "time of transition" where "stability in the US-China relationship is essential."
Sullivan added however that Biden was "not a conduit" for any backchannel messages from Xi to Trump about the shape of future relations.
- Climate fears -
With Biden making his swan song on the international stage, Sullivan said the president had also taken time for some informal chat with Xi.
The highly anticipated meeting came after a two-day summit of the 21-member Asia-Pacific Economic Cooperation (APEC) grouping that Xi and Biden had both attended.
Before their encounter, the men had taken part in a closed-door "retreat" with other APEC leaders from countries including Canada, Chile, Singapore, Australia, Malaysia and Japan.
Russia was represented by a deputy prime minister.
Billionaire Trump's comeback cast a cloud of uncertainty over the meeting -- and will do the same over a meeting of G20 leaders in Rio de Janeiro next week.
The Chinese president reiterated concerns earlier Saturday about mounting "protectionism" -- another apparent swipe at Trump's "America First" policies -- and urged APEC nations to "unite and cooperate."
Biden meanwhile spent the APEC summit trying to shore up relations with key US allies including Japan and South Korea ahead of Trump's return.
The US president-elect is also casting a pall over Biden's policies on climate change, with his threats to leave international pacts and increase drilling for fossil fuels.
Biden will become the first sitting US president to visit the Amazon rainforest when he stops in the Brazilian city of Manaus on Sunday on the way to Rio for the G20 meeting, where Xi is also headed.
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