Washington (CNN) Former UK Prime Minister Tony Blair denied a report that he had floated to White House adviser Jared Kushner the possibility that British intelligence could have had Trump campaign staff or then-candidate Trump under surveillance.

The assertion, made in a new book by Michael Wolff, said that Blair, "seeking to prove his usefulness" to the White House, told the administration last year that surveillance by the British could have taken place.

Blair has denied having such a conversation, and the book, "Fire and Fury," says the CIA also denied it. National Security Agency Director Michael Rogers denied a similar claim in a congressional hearing.

"I have never had such conversations in the White House or outside of the White House, with Jared Kushner or with anybody else," Blair told the BBC

The White House has slammed the book, with press secretary Sarah Sanders calling it "complete fantasy," and an attorney for Trump sending a legal threat to the book's author and publisher.

CNN has not independently confirmed Wolff's assertions.

The book says Blair visited Trump son-in-law and adviser Jared Kushner last February, when he suggested "the possibility that the British had had the Trump campaign staff under surveillance, monitoring its telephone calls and other communications and possibly even Trump himself."

The book adds: "It's unclear whether Blair's information was rumor, informed conjecture, his own speculation, or solid stuff. But, as it churned and festered in the President's mind, Kushner and Bannon went out to CIA headquarters in Langley to meet with (CIA Director) Mike Pompeo and his deputy director Gina Haspel to check it out. A few days later, the CIA opaquely reported back that the the information was not correct; it was a 'miscommunication.'"

The book said Trump repeatedly referenced the situation in his "nighttime calls" and kept coming back to what Blair and others had said.

The UK intelligence agency GCHQ -- the equivalent of the US National Security Agency -- reiterated its denial of any spying claims.

"As we have previously stated, allegations that GCHQ was asked to conduct 'wire-tapping' against Trump Tower are nonsense. They are utterly ridiculous and should be ignored," a GCHQ spokesperson said on Thursday.

Trump asserted without evidence last year that then-President Barack Obama had placed Trump Tower under surveillance.

"Terrible! Just found out that Obama had my 'wires tapped' in Trump Tower just before the victory. Nothing found. This is McCarthyism!" Trump said last March.

Terrible! Just found out that Obama had my "wires tapped" in Trump Tower just before the victory. Nothing found. This is McCarthyism! — Donald J. Trump (@realDonaldTrump) March 4, 2017

Then-White House press secretary Sean Spicer later cited a Fox News report alleging UK intelligence spied on Trump at Obama's behest, which GCHQ at the time called "nonsense."

The Justice Department said last September that there was no evidence to support Trump's wiretapping claim, though CNN reported later that month that US investigators had wiretapped former Trump campaign chairman Paul Manafort before and after the election, including a period when Manafort was known to talk to Trump.