"We are rounding the turn".
"We have to make sure there's enforcement", Mr Trump said.
A deal would allow a summit between him and Chinese President Xi Jinping, he said.
The damaging trade war has cast a shadow over global trade and is contributing to a slowdown in the world economy, which has recently spooked financial markets.
The stock market's slide late previous year increased the influence of market-friendly figures such as Treasury Secretary Steven Mnuchin, a former executive at the investment bank Goldman Sachs.
"People say we have the best relationship of any president-president, because he's called president also", Trump said of Xi in a Fox News interview at the time.
Earlier, Trump said that the negotiations were for an agreement covering the full range of United States concerns.
The progress on US-China trade was enough to keep the safe-haven yen under pressure as the dollar reached its highest in three weeks at $111.79, bouncing back down from Wednesday's low of $96.962. "If we have a deal, then we're going to have a summit", Trump said.
Goods trade between the US and China, the world's two largest economies, totaled $660bn a year ago, according to US Census Bureau data, consisting of imports of $540bn from China and $120bn in exports to China.More news: Oil Rally Slows On Surprise Crude Build
Meanwhile, a study revealed that the burden of U.S tariffs on Chinese imports is hurting the American economy contrary to what Trump is claiming. "We'll tariff their cars", he said.
Of course Trump is also right that all those nations would benefit from spending less on their military, particularly the United States, which is spending well beyond anyone else's level keeping up a military intervening across the entire planet. If the two leaders decide to haggle in person over any final details, a third-country venue is more likely.
Trump said there was potentially for an "epic" and "historic" agreement.
Despite Trump's rosy gloss on the talks on Thursday, US Trade Representative Robert Lighthizer told reporters that major issues were left to be resolved.
Trump has come under fire for using trade deficits as shorthand for who is "winning" in trade, but no serious economist sees things that way. In retaliation, China has targeted $110bn in American imports.
The US has already imposed tariffs on $250bn worth of Chinese goods, and China has retaliated with duties on $110bn of US products.
Lighthizer, the chief U.S. trade negotiator, also has complained about Chinese hackers stealing American trade secrets and discriminating against foreign companies.
Boone warned tensions between the USA and China would most likely continue well beyond an expected trade deal over the coming weeks.
Liu met with the president after two days of talks between Chinese and American trade negotiators in Washington. China is the biggest buyer of USA soybeans, the bulk of which are genetically modified. "Those issues are critical: Manufacturers in the United States have always been harmed by China's unfair trade practices".