Bloomberg cited unnamed sources as saying that Trump had also asked his team to begin drafting terms for an agreement on trade with China, although economic adviser Larry Kudlow later denied that claim.
"If a Trump-Xi deal at the G20 Summit can set the broad framework for renewed bilateral trade negotiations, it may help the global trading system avert the abyss of an escalating US-China trade war in 2019", said Rajiv Biswas, Asia Pacific chief economist at IHS Markit in Singapore. At a campaign rally in Columbia, Missouri, he added: "They want to make a deal".
But there remains plenty of reason for pessimism.
Tu Xinquan, a trade expert at Beijing's University of International Business and Economics, said he was sceptical that the Trump administration was truly ready to deal, and that other factors, such as the mid-term congressional elections, could be driving statements from the administration. And we're doing very well. Xi said later to be willing to meet Trump.
Tensions have increased in recent months with USA officials accusing China of meddling in the United States midterm elections, conducting cyber espionage and military aggression. His critics say the trade pacts that he has so far completed, including a renegotiation of Nafta, represent little more than incremental improvements.
The U.S. traditionally imports more good than it exports because Americans consume more than they produce relative to the rest of the world.
Earlier Friday, the Chinese foreign ministry said Trump and Xi had an "extremely positive" phone conversation on Thursday.
"I think we'll make a deal with China and I think it'll be a very fair deal for everybody, but it will be a good deal for the United States", Trump said.
The economic slowdown, coupled with Beijing's dispute with Trump, has fed gloom among investors.
President Trump said he will discuss trade with President Xi when the 2 meet for dinner on the sidelines of the G-20 leaders' Summit at the end of November in Buenos Aires, Argentina.
Trump revoked his government's support for a joint statement following the G7 conference in Canada after Canadian Prime Minister Justin Trudeau criticized USA steel and aluminum tariffs.
Tennessee Senator Lamar Alexander told the official the USA and China are competitors not adversaries and suggested they can prosper together.
Trump said a deal with China would also be good for Beijing.
Much remains unclear about Trump's posture toward China.
China and the United States are locked in an increasingly bitter trade war, and have placed tariffs on some of each others' imports.
The Trump administration has been imposing tariffs on various countries, including China, claiming that the U.S. was "being treated unfairly in trade" and accusing Beijing of "stealing intellectual property". Japan's Nikkei index followed with a 2.4 percent rise and USA stock futures increased 0.7 percent, Reuters reported on Friday.
Hong Kong surged more than two per cent and Shanghai jumped more than one per cent.