The five-page transcript of a conversation between Bush and British Prime Minister Tony Blair reveals that Blair talked Bush out of launching a military strike on the station, unnamed sources told the daily.
The transcript of the pair's talks during Blair's 16 April 2004 visit to Washington allegedly shows Bush wanted to attack the satellite channel's headquarters in Doha, Qatar.
Blair allegedly feared such a strike, in the capital of Qatar, a key Western ally in the Gulf, would spark revenge attacks.
Aljazeera has said in a news statement that it is investigating the report and urged the US and British governments to challenge it.
A British civil servant has been charged under the Official Secrets Act for allegedly leaking the government memo.
Civil servant accused
Cabinet Office civil servant David Keogh is accused of passing the memo to Leo O'Connor, who formerly worked for former British lawmaker Tony Clarke.
Both Keogh and O'Connor are scheduled to appear at London's Bow Street Magistrates Court next week.
"This doesn't surprise me. Whenever there is truth - unpleasant truth - it seems this administration wants to bomb it."
"You guys have reached a new low in shamelessness. The Daily Mirror? ...It is a rag, filled with lies and half truths."
According to the Daily Mirror, Clarke returned the memo to Blair's office. Clarke could not immediately be contacted for comment on Tuesday.
The Mirror on Tuesday quoted an unnamed British government official as saying Bush's threat was "humorous, not serious".
Aljazeera's coverage of the war in Iraq had drawn criticism from Washington after the US-led March 2003 invasion.
A source told the Mirror: "The memo is explosive and hugely damaging to Bush.
"He made clear he wanted to bomb Aljazeera in Qatar and elsewhere. Blair replied that would cause a big problem.
"There's no doubt what Bush wanted to do - and no doubt Blair didn't want him to do it."
Another source said: "Bush was deadly serious, as was Blair. That much is absolutely clear from the language used by both men."
A spokesman for Blair's Downing Street office said: "We have got nothing to say about this story. We don't comment on leaked documents."
The US has criticised Aljazeera's
coverage of the Iraq war
Clarke, the former lawmaker, told Britain's domestic Press Association news agency that O'Connor had done "exactly the right thing" in bringing it to his attention.
The Mirror said such a strike would have been "the most spectacular foreign-policy disaster since the Iraq war itself".
The newspaper said that the memo "casts fresh doubt on claims that other attacks on Aljazeera were accidents". It cited the 2001 direct hit on the channel's Kabul office in Afghanistan.
In April 2003, an Aljazeera journalist died when its Baghdad office was struck during a US bombing campaign. Nabil Khoury, a US State Department spokesman in Doha, said the strike was a mistake.
In November 2002, Aljazeera's office in Kabul, Afghanistan, was destroyed by a US missile. None of the crew was at the office at the time. US officials said they believed the target was a terrorist site and did not know it was Aljazeera's office.
Downing Street challenged
Blair's former defence minister Peter Kilfoyle challenged Downing Street to publish the transcript.
Blair is said to have talked Bush
out of any attack on Aljazeera
"I believe that Downing Street ought to publish this memo in the interests of transparency, given that much of the detail appears to be in the public domain," Kilfoyle told the Press Association.
"I think they ought to clarify what exactly happened on this occasion. If it was the case that President Bush wanted to bomb Aljazeera in what is after all a friendly country, it speaks volumes and it raises questions about subsequent attacks that took place on the press that wasn't embedded with coalition forces."
Liberal Democrat foreign affairs spokesman Sir Menzies Campbell told the Press Association: "If true, then this underlines the desperation of the Bush administration as events in Iraq began to spiral out of control.
"On this occasion, the prime minister may have been successful in averting political disaster, but it shows how dangerous his relationship with President Bush has been."
Abd al-Bari Atwan's reaction
Speaking to Aljazeera from London on Monday, Abd al-Bari Atwan, chief editor of the London-based Al-Quds al-Arabi newspaper, said: "The issue of Bush's plan to bomb Aljazeera's headquarters in Doha will be widely discussed in Washington and London.
"Reporters in the US and Britain are enraged by reported US plans to use force against media organs.
"Arab and international media organs are now under a terrorist campaign launched by the US as it does not want the truth to be revealed.
Aljazeera offices in Kabul and
Baghdad were hit by US missiles
"This [US] administration has been disgraced as it has used immoral and illegal ways to occupy and tear out a country, kill more than 100,000 and wound more than 400,000 of its people.
"The results of the war, being revealed now in Iraq, have forced reporters to ask why they have been misled.
"New York Times has apologised, saying it has misled public opinion when it did not accurately investigate the objectives of the US administration.
"I believe that considering use force against a media station is the worst kind of media terrorism practised by a country which pretends to lead the free world, democratic values and media freedom."