Speaking to reporters on board Air Force One on Friday, the US president said: "We'll sue them for anywhere between one billion dollars ($A1.5 billion) and five billion dollars ($A7.7 billion), probably sometime next week."
The BBC apologised but said it refused to pay financial compensation.
The corporation said the splicing of the speech was an "error of judgment".
Still, it rejected his compensation demands, after Trump's lawyers threatened to sue the BBC for one billion dollars in damages unless a retraction and apology were published.
Mr Trump's lawyers had given the BBC a 10pm Friday deadline to respond.
Earlier in the week, Trump said in an interview with Fox News that the BBC had "defrauded the public" over the edit, which made it appear as if he was explicitly urging people to attack the US Capitol on January 6, 2021.
Separately, Trump told GB News in an interview broadcast on Saturday he felt an "obligation" to take legal action against the public broadcaster.
"I'm not looking to get into lawsuits, but I think I have an obligation to do it," he said.
"This was so egregious. If you don't do it, you don't stop it from happening again with other people."
Chairman Samir Shah has sent a personal letter to the White House to apologise for the editing, and lawyers for the corporation have written to the president's legal team, a BBC spokesperson said.
The spokesperson added: "While the BBC sincerely regrets the manner in which the video clip was edited, we strongly disagree there is a basis for a defamation claim."
The Panorama scandal prompted the resignations of two of the BBC's most senior executives: director-general Tim Davie and news chief Deborah Turness.
The program, broadcast a week before the 2024 US election results, spliced two clips together so that Mr Trump appeared to tell the crowd: "We're going to walk down to the Capitol ... and I'll be there with you. And we fight. We fight like hell."
The broadcaster said it will not air the Panorama episode Trump: A Second Chance? again, and published a retraction on the show's webpage on Thursday.
It said: "This program was reviewed after criticism of how President Donald Trump's 6th January 2021 speech was edited.
"During that sequence, we showed excerpts taken from different parts of the speech.
"However, we accept that our edit unintentionally created the impression that we were showing a single continuous section of the speech, rather than excerpts from different points in the speech, and that this gave the mistaken impression that President Trump had made a direct call for violent action.
"The BBC would like to apologise to President Trump for that error of judgment."
A spokesperson added: "The BBC has no plans to rebroadcast the documentary Trump: A Second Chance? on any BBC platforms."
On Thursday, reports said that the BBC faced separate accusations of misleading viewers about Trump's 2021 Capitol speech more than two years before the Panorama edit aired.
In an episode broadcast in June 2022, Newsnight reportedly played an edited version of his speech, similar to the one used in the Panorama program.
A BBC spokesperson said about the fresh claims, reported by The Telegraph's Daily T podcast: "The BBC holds itself to the highest editorial standards. This matter has been brought to our attention and we are now looking into it."
16°C