Lawyers for the BBC argued that Ivanka Trump, her husband Jared Kushner, and Donald Trump Jr have "personal knowledge" and likely have records relevant to elements of Trump's claims against the broadcaster, according to a court document filed on Friday in the Federal Court in Florida.
The broadcaster has been unable to serve subpoenas because the three have Secret Service protection and other security personnel around them, the document says.
Trump filed the lawsuit in December seeking $US10 billion ($A14 billion) in damages from the BBC, accusing it of defamation as well as deceptive and unfair trade practices.
The allegations centre on the way a 2024 documentary edited a speech Trump gave on January 6, 2021, before protesters attacked the Capitol in Washington.
The lawsuit accuses the BBC of "splicing together two entirely separate parts of President Trump's speech" to "intentionally misrepresent the meaning of what President Trump said".
It added that the editing was "a brazen attempt to interfere in and influence" the 2024 US presidential election.
The BBC has apologised to Trump for the misleading edit, but said it had not defamed him.
The latest court filing said that Trump's family members had knowledge of issues that were relevant to Trump's argument "that it would be materially false to imply that he incited violence" on January 6, 2021.
It cited that Donald Trump Jr and Ivanka Trump were both present in the Oval Office when Trump was still revising his speech, adding that Donald Trump Jr spoke directly with his father after violence had broken out at the Capitol.
The filing says the BBC asked Trump to accept the subpoenas on his daughter and son-in-law's behalf or to direct the Secret Service to allow the subpoenas to be served, but he refused. The broadcaster asked the court to allow the subpoenas to be served by email and certified mail.
The judge has given Trump's legal team until Friday to respond to the BBC's request.
Trump's legal team said that "the BBC is simply trying to distract away from their own obvious liability".
"The BBC intentionally defamed President Donald J Trump, and now the BBC is seeking to harass him, his family, and supporters by abusing the deposition process," according to the team.
A trial has been provisionally set for February in the case.
Last week, a federal judge in Florida granted Trump a temporary reprieve from an order that he provide details of his business empire's financial performance as part of the lawsuit.
US District Judge Roy Altman agreed to put that order on hold while he considers an amended complaint from Trump that would narrow his claim that he suffered damage to both business and reputation.
