Trump's comments on the China trade talks helped fuel a broad rally in the US stock market, along with the president's comments that he did not anticipate another government shutdown despite not being "happy" with a tentative congressional deal for border security funding.
One of Trump's most persistent economic promises has been to rewrite the US relationship with China.
The U.S. delegation also included Treasury Secretary Steven Mnuchin and David Malpass, a Treasury undersecretary who is Trump's nominee for World Bank president.
The first source added that the host country would hold a banquet for the USA delegation in a Chinese restaurant, and Chinese Vice Premier Liu He would attend the event.
A report in the South China Morning Post said Xi would personally meet the U.S. delegation in Beijing, suggesting a redoubled effort to make progress on a deal.
US tariffs on $200 billion worth of imports from China are scheduled to rise to 25 percent from 10 percent if the two sides can not reach a deal by a March 1 deadline, increasing pain and costs in sectors from consumer electronics to agriculture. It will be his second meeting with Lighthizer following last month's talks in Washington.
Sentiment was buoyed by developments on trade and hopes of avoiding another USA government shutdown.
The Chinese foreign ministry on Monday said the People's Liberation Army Navy sent vessels to track the USA warships sailing near the Spratlys and warned them to leave the area.
The U.S. president had said on Tuesday that the deadline for an agreement could "slide for a little while", but he preferred not to do so.
Mr Trump has also threatened further tariffs on an additional $267bn worth of Chinese products.
Beijing has tried to deflect pressure by emphasizing China's growth as an export market.
Donald Trump said the new tariff would kick in after March 1 in the absence of a trade deal.
Regulators have announced plans to improve protection of foreign patents and copyrights.
Both the Chinese government and Huawei have dismissed these concerns. "President Trump and President Xi sitting down face-to-face figuring that out and getting that final deal because they are the only two that'll ultimately be able to nail that down", Sanders said.
In addition, a banquet would be hosted for the U.S. delegation in "a Chinese cuisine restaurant" in downtown Beijing later this week, with Chinese Vice-Premier Liu He expected to toast the United States delegates, the report added.
The Hong Kong newspaper The South China Morning Post reported Monday the meeting could take place in late March on the southern Chinese island of Hainan.
Trump's comments are at odds with the firm line held by other members of the administration until now.
"We certainly hope not", said Waterman.