"Your leader is coming over to see me very soon," Mr Trump told ABC reporter John Lyons while speaking to journalists at the White House before leaving for the United Kingdom.
Mr Trump did not mention when or where he would meet Mr Albanese, and the federal government has yet to confirm the meeting.
The president's comments came after he objected to questions from Mr Lyons about whether it was appropriate for a US president to be "engaged in so much business activity" while in office.
Mr Trump said his children "are running the business".
"In my opinion, you are hurting Australia very much right now," the president added.
"And they want to get along with me.
"You know, your leader is coming over to see me very soon.
"I'm going to tell him about you.
"You set a very bad tone."
Mr Lyons, the ABC's Americas Editor and an award-winning journalist, defended his line of questioning.
"Our job as journalists is to ask questions that the average person would be interested in," he told ABC television.
"And I think the average person in Australia would be interested in 'how is a president becoming so wealthy in office'.
"We asked them politely, respectfully. They were not shouted. They were not abusive."
A social media account linked to the White House said Mr Trump had smacked "down a rude foreign Fake News loser".
Federal MPs from across the political divide defended the ABC, saying journalists had the right to ask difficult questions.
"Donald Trump got asked some of those tough questions and it's something that we see every day in the Australian media," minister Clare O'Neil told Seven's Sunrise program on Wednesday.
"The journalists are there to try to keep politicians accountable and they're entitled to ask difficult questions."
Nationals senator Bridget McKenzie agreed, adding that it was "the scoop of the year for the ABC" because it was able to confirm that a meeting was forthcoming.
"There's nothing wrong with journalists asking tough questions," she said.
Asked if she could confirm the meeting, Ms O'Neil said "it's certainly what's been intended and we've got two incredibly busy people here".
But Liberal senator and former ABC journalist Sarah Henderson said the national broadcaster needed to explain itself.
"At a time when trade, defence and national security are such crucial issues in our relationship with our closest ally, it would be helpful if the ABC could explain this line of questioning," she said on X.
"Australians should expect the highest standards of our publicly funded national broadcaster."
The prime minister is preparing to travel to New York in the coming days for the United Nations General Assembly meeting in New York, where a face-to-face with Mr Trump on the sidelines is a possibility.
Mr Trump praised Mr Albanese as a "good man" after the two leaders held their fourth one-on-one phone call earlier in September.
The prime minister described his call with Mr Trump as "really warm".
A face-to-face meeting between the two leaders had been planned on the sidelines of the G7 summit in Canada in June but was cancelled after Mr Trump returned to the US early to deal with the Iran-Israel conflict.
The UN's "high-level week" starts in New York on Monday.
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