He is said to have been "angling" for a post-election Middle East adviser role.
Mr Blair's office said the allegations "are a complete fabrication".
A spokeswoman added they "have no basis in reality and are simply untrue".
She is quoted in the Times, which first reported the story, as saying Mr Blair made no pitch to be Mr Trump's Middle East envoy.
The account of an alleged meeting between Mr Blair, Mr Kushner and a senior aide at the White House in February 2017, the month after the presidential inauguration, is contained in the book by journalist Michael Wolff and reported in the Times.
According to Mr Wolff's account, in Fire and Fury: Inside the Trump White House, Mr Blair shared the rumour that, during the election, British intelligence "had had the Trump campaign staff under surveillance, monitoring its telephone calls and other communications".
The former PM is reported to have given the impression that Barack Obama's administration had hinted that such surveillance would be helpful.
Last March the US TV channel Fox News carried an unsubstantiated claim that UK intelligence services could have been involved in an alleged spying operation on Trump Tower in New York.
The allegation was repeated by the then-White House press secretary Sean Spicer, prompting a rare on-the-record response from Britain's intelligence chiefs.
The UK's spy agency GCHQ said the claims "should be ignored, describing them as "nonsense" and "utterly ridiculous".
Meanwhile, Mr Trump has disavowed his former aide Steve Bannon, saying he "lost his mind" after he lost his job.
It follows reported comments from Mr Bannon in the book in which he describes a meeting between the president's son, Donald Trump Jr, and a group of Russians, as "treasonous" and "unpatriotic".
The Senate, House of Representatives and a US Department of Justice team led by special counsel Robert Mueller are investigating alleged Russian interference in the presidential election and alleged attempts to undermine Hillary Clinton.
The claims have been denied both by the Kremlin and Mr Trump.
The president has also vehemently denied any collusion with Russia.
On Twitter, Mr Bannon described the book's claims about him as "total fake news and taken out of context".
Responding to Mr Trump's comments about him, he tweeted: "I won't speak about who lost their minds or ever had one. But it's easy to see who will be played by the Opposition Party. We should all fight for the common goals and ideas we have."
Mr Trump's lawyers have sent a cease-and-desist letter to Mr Bannon, US media reports said.
On Wednesday White House spokeswoman Sarah Sanders described Mr Wolff's book as "filled with false and misleading accounts from individuals who have no access or influence with the White House".
"Participating in a book that can only be described as a trashy tabloid fiction exposes their sad desperate attempts at relevancy," she said.
It's not unusual for former political allies to fall out. It happens all the time. But normally it's done in private, and maybe hinted at in public. Not this time. Not with this President.
This is a vicious knife fight between Donald Trump and his former campaign chief and White House Chief strategist being played out across social media.
Other Trump family members and supporters are piling in too. Why? Because Steve Bannon has enraged Donald Trump with his description of a meeting that took place between the president's son, Don Jr, and son in law, Jared Kushner, with a Kremlin-linked, Russian lawyer as unpatriotic and treasonous.
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