The administration has scheduled talks with South Korea and Japan, two close allies and major trading partners, and Italian Prime Minister Giorgia Meloni is due to visit next week.
"These are tailored, highly tailored deals," Trump said at a White House event, where he signed executive orders aimed at boosting coal production. "We've had talks with many, many countries, over 70, they all want to come in. Our problem is, can't see that many that fast."
But the White House made clear that country-specific tariffs of up to 50% would nevertheless take effect at 12:01 a.m. Eastern Time (0401 GMT), as planned.
Those tariffs will be especially steep for China, as Trump has ratcheted up duties on its imports to 104% in response to countertariffs Beijing announced last week. China has refused to bow to what it called blackmail and has vowed to fight to the end.
Administration officials said they would not prioritize negotiations with the world's No. 2 economic power.
"Right now, we've received the instruction to prioritize our allies and our trading partners like Japan and Korea and others," White House economic adviser Kevin Hassett said on Fox News.
At an evening speech to Republican lawmakers, Trump said he would soon announce "major" tariffs on pharmaceutical imports, arguing the duties would push drug companies to move manufacturing operations to the U.S.
Trump's sweeping tariffs have raised fears of recession and upended a global trading order that has been in place for decades.