Xi's comments in a phone call with Trump follow Beijing's displeasure over U.S. arms sales to rival Taiwan, U.S. sanctions against a Chinese bank over its dealings with North Korea and, most recently, the sailing of a U.S. destroyer within the territorial seas limit of a Chinese-claimed island in the South China Sea.
The expectation for China and Germany to mount an informal alliance to take up the leadership role comes amid accusations by some countries that the USA under the Trump administration has abdicated its world leadership role.
"We've had similar dynamics under Bush, Clinton and Obama", former China adviser to President Barack Obama, Jeffrey A. Bader, said.
The statement said Trump also expressed his desire "to seek more balanced trade relations" with America's trading partners.
Trump instructed Tillerson and Mattis to go ahead with measures that had been placed on hold pending a renewed effort by China to get tough with North Korea.
Just two days ago, Trump declared U.S. patience with North Korea was "over".
Among those topics were containing the Democratic People's Republic of Korea, as well as the United States' naval presence in the South China Sea.
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John Cammack was left with severe burns after falling from a fire engine when the winds shifted and the flames changed direction. In Arizona, Governor Doug Ducey declared a state of emergency on Friday as more than 20 wildfires burned in the state.
The series of moves have marked a sharp reversal from the friendly tone struck by Trump since his meeting with Xi at the USA president's Mar-a-Lago resort in Florida in April.
China hopes that the US will handle the Taiwan issue appropriately in accordance with the "One China" principle and the three China-US joint communiques, Xi said.
Relations between the USA and China are back in rough waters ahead of the G20 Summit. Previous US administrations have sold arms to Taiwan supposedly to ensure against a Chinese attack. In a separate, but clearly related, move the USA has announced a $1.4bn arms deal with Taiwan, to which Beijing has reacted angrily. That was in April, at a meeting at the Trump estate in Florida.
Beijing was given another cause for concern after the Senate Armed Services Committee approved a change in U.S. policy toward Taiwan, to authorise regular visits by USA warships to Taiwanese ports and allow Taiwanese naval vessels to visit United States bases in the Pacific. The legislation also calls for the USA to develop Taiwan's "undersea warfare capabilities" and strengthen strategic cooperation with Taiwan.
Trump might also have heard Xi's complaints about US Treasury Department actions, namely the leveling of "secondary sanctions" on China's small but crucial Bank of Dandong.
Washington is also said to be preparing additional sanctions against other Chinese companies and those could be announced over the next few weeks.
The latest irritant came on Sunday when the USS Stethem destroyer sailed less than 12 nautical miles from tiny Triton Island in the Paracel Islands archipelago, which is claimed by China as well as Taiwan and Vietnam, a USA official told AFP. The action taken by the warship, the USS Stethem, off Triton Island in the Paracel Islands prompted a furious response from the Chinese government, which called it a "serious political and military provocation".
China is responsible for 90% of North Korean trade and key to the robustness of latest United Nations sanctions.
Trump has been weighing new quotas or tariffs on steel imports for national security reasons and plans to discuss his concerns at the G20.
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