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.
The impetus for Trump's request to delay "appears to be the flat refusal by the Donald J Trump Revocable Trust ... to provide any financial information under subpoena", the newspaper cited the BBC as saying in a court filing.
That action came despite Trump's claims that the BBC injured "the value of his brand, properties, and businesses" and the president's "own refusal, to date, to provide any financial information in discovery", the paper said.
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.
The broadcaster and the White House did not immediately respond to requests for comment.
A spokesperson for Trump's legal team told the newspaper 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 documentary's financial impact, 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 BBC legal team has sought financial documents that reflect the trust's holdings and value, assets, inventories and properties, the paper said, citing court filings from May that it viewed.
The request covers almost 400 entities owned by or associated with the trust, as well as requests for tax returns, 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.
