Trump has accused the publicly funded broadcaster of defaming him by splicing together parts of a January 6, 2021, speech to make it appear that he directed supporters to storm the US Capitol.
Trump's legal team said Magistrate Judge Enjolique Lett's "prior representation of a client that was directly adverse to President Trump creates an appearance of impropriety", in a request opposed by the BBC.
"The impetus ... appears to be the flat refusal by the Donald J Trump Revocable Trust, a third party that is represented by the same counsel as plaintiff, to provide any financial information under subpoena," the BBC's lawyers said.
Trump's lawyers, however, said they sought only a brief stay until the reassignment request was decided.
It was not immediately clear when such a ruling would be made.
The broadcaster and the White House did not immediately respond to requests for comment.
Trump's lawsuit, filed in Florida in December, says the BBC violated a state law that bars deceptive and unfair trade practices.
He is seeking damages of at least $US5 billion on each of its two counts.
A spokesperson for Trump's legal team told the Financial Times, which first reported the news on Friday, that the BBC was liable for "intentionally and maliciously defaming him by distorting and manipulating his speech".
In a statement, the spokesperson said: "President Trump will continue to hold accountable the BBC and all those who traffic in fake news."
In its bid to ascertain the financial impact of the film, the BBC has subpoenaed the trust, managed by Trump's eldest son, Donald Trump Jr, as the sole trustee, which holds the president's business interests and assets, the paper said.
The documentary, first broadcast in 2024 shortly before a presidential election Trump won, featured a section in which he told supporters to march on the Capitol and another, from nearly an hour later, where he said, "Fight like hell."
The BBC apologised to Trump for the edit, but wants his lawsuit thrown out. Trump's subsequent re-election showed the alleged defamation did not harm his reputation, the broadcaster said in court papers released in March.
