The US president spoke to Xi on Thursday and later tweeted that trade talks with China were "moving along nicely" ahead of face-to-face talks between the pair at the G20 summit in Argentina later this month.
"I hope that China and the USA will meet each other halfway and will appropriately handle their differences with mutual respect and equal cooperation", stated the Chinese premier.
Still, investors cheered the resumption of dialogue and a report that Trump was taking steps to resolve the tariff war, with shares in Asia hitting three-week highs on today and the dollar softening.
Separately, the South China Morning Post - also citing unnamed sources - reported that Mr Trump had offered to host a dinner for the Chinese president after the G20 summit.
Mr Trump has asked key U.S. officials to begin drafting potential terms, according to the report.
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"China wants to make a deal".
In comments in state media, Xi said he hoped China and the United States would be able to promote a steady and healthy relationship, and that he was willing to meet Trump in Argentina.
"The economic teams of the two countries should strengthen contact, conduct consultations on issues of mutual concern, and push for a mutually acceptable solution to China-US economic and trade issues", Xi said.
Xi was quoted by CCTV as saying after the call with Trump that the two leaders had hoped to expand bilateral trade cooperation.
Neither leader specified any details of possible progress in their first known direct discussion in several months.
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The onshore yuan was headed for its biggest two-day advance in more than 11 years amid prospects that China and the US are working toward a possible deal in a trade war that has roiled global financial markets.
Trump said he also talked with Xi about "many subjects", including a "good discussion" on North Korea, where Trump is trying to negotiate an end to Pyongyang's nuclear weapons program.
"What negotiations have lacked to date has been that (presidential) level of engagement".
Tai Hui at JPMorgan Asset Management said a stabilising trade relationship between USA and China and more stimulus from Beijing would be the key ingredients to revive market confidence in Asia.
'We're going way up, and China has been hurt over the last number of months. He said further meetings with Xi are being scheduled for the G-20 summit of the world's biggest economies, set for November 30 in Buenos Aires, the first ever such summit in South America. But Kudlow cautioned that Trump would "aggressively" pursue his agenda against China, if no deals were reached on intellectual property theft, cybersecurity and tariffs on commodities, among other issues.
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