Earlier the US national intelligence director's office said it was still investigating how the virus began.
But the office said it had determined Covid-19 "was not manmade or genetically modified".
China has rejected the lab theory and criticised the US response to Covid-19.
Since emerging in the Chinese city of Wuhan at the end of last year, the coronavirus is confirmed to have infected 3.2 million people worldwide and killed more than 230,000.
What did President Trump say?
At the White House on Thursday, Mr Trump was asked by a reporter: "Have you seen anything at this point that gives you a high degree of confidence that the Wuhan Institute of Virology was the origin of this virus?"
"Yes, I have. Yes, I have," said the president, without specifying. "And I think the World Health Organization should be ashamed of themselves because they're like the public relations agency for China."
Asked later to clarify his comment, he said: "I can't tell you that. I'm not allowed to tell you that."
He also told reporters: "Whether they [China] made a mistake, or whether it started off as a mistake and then they made another one, or did somebody do something on purpose?
"I don't understand how traffic, how people weren't allowed into the rest of China, but they were allowed into the rest of the world. That's a bad, that's a hard question for them to answer."
The New York Times reported on Thursday that senior White House officials have asked the US intelligence community to investigate whether the virus came from a Wuhan research laboratory.
Intelligence agencies have also been tasked with determining if China and the WHO withheld information about the virus early on, unnamed officials told NBC News on Wednesday.
What did the intelligence chief say?
In a rare public statement, the Office of the Director of National Intelligence, which oversees US spy agencies, said on Thursday it concurs with the "wide scientific consensus" regarding Covid-19's natural origins.
"The [intelligence community] will continue to rigorously examine emerging information and intelligence to determine whether the outbreak began through contact with infected animals or if it was the result of an accident at a laboratory in Wuhan."
It was the first clear response from American intelligence debunking conspiracy theories - both from the US and China - that the virus is a bioweapon.
What's the background?
Mr Trump has recently been escalating his war of words with China over the pandemic after what officials within the US president's administration had described as a truce with Beijing.
On Wednesday, he suggested China wanted him to lose his re-election bid in November.
Mr Trump has formerly accused Chinese officials of covering up the virus early on and saying they could have stopped the disease from spreading.
He has similarly criticised the WHO and withdrawn US funding for the global body.
China's Foreign Ministry, meanwhile, has accused the Trump administration of trying to distract from its own problems tackling the crisis.
A ministry spokesman has also repeatedly promoted the idea - without evidence - that Covid-19 might have originated in the US.
According to the Washington Post, the Trump administration is looking into ways to punish China financially. Discussions reportedly include allowing the US government to sue China for damages or cancelling debt obligations.
The US-China propaganda war
This is the first definitive statement on the matter from US intelligence agencies. It rejects the most extreme of the conspiracy theories about the pandemic's origins - that the Chinese developed and unleashed the coronavirus as a bioweapon.
But it doesn't rule out the possibility that the virus was accidentally leaked from a Wuhan laboratory studying infectious diseases.
US Secretary of State Mike Pompeo in particular has talked up that scenario, urging China to let outside experts into the facility, and raising questions about lab safety in other parts of the country. The Chinese government says any such allegations are unfounded and fabricated out of nothing.
Claims and counterclaims about the origins of the outbreak are part of a propaganda war over China's handling of the coronavirus crisis.
But they also reflect US frustration with the Chinese for not sharing more data about how the pandemic developed. BBC