A senior al-Jazeera executive is in the UK to demand publication of a memo in which George Bush allegedly discusses bombing the TV station's HQ.
Al-Jazeera journalists stage a protest over the memo at their Qatar HQ
Wadah Khanfar, al-Jazeera's director general, is hoping to meet UK government officials to press its case.
A spokesman for al-Jazeera told the BBC News website that the channel only wanted the record set straight.
Downing Street said: "We are quite happy to talk to al-Jazeera as we are to other broadcasters."
The Italian La Stampa newspaper has reported that Mr Khanfar had "demanded an urgent meeting" with UK Prime Minister Tony Blair.
Mr Khanfar told La Stampa: "We want to know whether Bush really did want to attack al-Jazeera last year and was dissuaded from doing so by the British prime minister, as the British press has claimed.
"We will be silent only when we get the truth."
According to press reports, the memo includes a transcript record of Tony Blair attempting to persuade Mr Bush not to take military action against the al-Jazeera headquarters.
The station is based in Qatar, a close ally of Washington and the location of US military headquarters during the Iraq war. The White House has dismissed reports of the conversation as "outlandish".
UK Attorney General Lord Goldsmith has warned media outlets not to publish the contents of the memo, citing the Official Secrets Act.
But Labour MP and former defence minister Peter Kilfoyle has tabled a Commons motion calling for it to be made public.
He said ministers wanted to keep it secret to save political embarrassment rather than protect national security.
In an early-day motion, Mr Kilfoyle said he was "appalled" by Mr Bush's reported comments and called on the government to publish the transcript.
Mr Blair's April 2004 visit to the White House took place in the wake of a failed US attempt to root out insurgents in the city of Falluja, during which 30 US Marines and an estimated 1,000 Iraqis died.
Mr Kilfoyle said he believed that plans for a further assault on Falluja may have been discussed at the meeting, and his motion voices concern at Mr Blair's failure to restrain the attack, which eventually took place in November last year.
Mr Kilfoyle said: "I am calling for the publication of the record, which we know exists, of a meeting in April 2004 in which George Bush allegedly called for the bombing of the al-Jazeera HQ in Qatar and which also discussed the launch of the assault on Falluja, which I find a very important issue, particularly related to the British attempts at restraint.
"I would hope we can have a fair and full discussion of the very important issues that were discussed at that meeting. "This is not about national security. It is about political embarrassment.
"The information is out there in the public domain and it seems ludicrous that the media can't discuss it in its entirety."
Cabinet Office civil servant David Keogh has been charged under the Official Secrets Act of passing the memo to former MP's researcher Leo O'Connor.
Both men are bailed to appear at Bow Street Magistrates Court next week.
Mr Khanfar, who is planning to spend "a few days" in the country, is due to take part in a debate with Daily Mirror journalist Kevin Maguire, who first published details of the memo.
BBC security correspondent Frank Gardner says this latest row follows a history of tension and mutual suspicion between al-Jazeera and the US administration.
Many of al-Jazeera's employees have long been privately convinced that their offices in Kabul and Baghdad were deliberately targeted by the Pentagon in 2001 and 2003 respectively.
US officials have accused al-Jazeera of being a mouthpiece for al-Qaeda and of being openly hostile to the US.