Their meeting started with smiles and a handshake despite Trump's desire to eradicate the US-Canada border, a prospect that has chilled bilateral relations.
The subject quickly came up as they took questions from reporters.
"We're not going to be discussing that unless somebody wants to discuss it," Trump said.
"It would really be a wonderful marriage."
Carney put down the idea firmly.
"It's not for sale, it won't be for sale - ever," he told Trump in the Oval Office.
"Never say never, never say never," Trump said.
Trump, whose tariff policy has rattled world markets, said he and Carney would discuss "tough points," an allusion to the president's belief that the United States can do without Canadian products.
"Regardless of anything, we're going to be friends with Canada," he said.
Carney's Liberal Party won the April 28 election on promises to tackle Trump and create a new bilateral economic and security relationship with the United States.
Shortly before Carney arrived, Trump posted a message on social media.
"I very much want to work with him, but cannot understand one simple TRUTH — Why is America subsidising Canada by $200 Billion Dollars a year, in addition to giving them FREE Military Protection, and many other things? We don't need their Cars, we don't need their Energy, we don't need their Lumber, we don't need ANYTHING they have, other than their friendship, which hopefully we will always maintain. They, on the other hand, need EVERYTHING from us!"
Trump appeared to be referring to the trade deficit the US has with Canada due mostly to US imports of Canadian oil, although Canada's merchandise trade surplus was C$102.3 billion ($A115.21 billion) in 2024.
Carney, a 60-year-old ex-central banker with no previous political experience, was elected Liberal leader in March to replace Justin Trudeau, who had a poor relationship with Trump.
Canada is the US' second-largest individual trading partner after Mexico, and the largest export market for US goods.
More than $US760 billion ($A1.2 trillion) in goods flowed between the two countries last year.
Ahead of the meeting, the US Commerce Department reported on Tuesday Canada's goods trade surplus with the US narrowed to a five-month low in March, the month when Trump's hefty tariffs on imported steel and aluminium took effect.
Canadian exports to the US plunged by $US3.7 billion, the second-largest drop on record.
Canadian data showed the drop in US exports was almost compensated by an increase to the rest of the world, as Canadian companies sought new markets.
Trump in March imposed a 25 per cent tariff on all steel and aluminium imports and then slapped another 25 per cent tariff on cars and parts that did not comply with a North American free trade agreement.
Trump showered his counterpart with compliments and radiated warmth at the start of the press conference, saying that "Canada chose a very talented person, a very good person".
with AP
