The Daily Mirror reports that Bush was dissuaded from the plan by British Prime Minister Tony Blair, who said that an attack on an ally would provoke a major backlash.
Quoting a Number 10 memo, which the Mirror said was marked "Top Secret", the newspaper said that Bush threatened to unleash "military action" against the network.
Sources told the newspaper that Bush made it clear that he wanted to bomb Al Jazeera in Qatar and other locations, and denied reports that the threat was made in a humorous way. The paper said that the plan was revealed by Bush to Blair at a White House summit on April 16 2004.
Al Jazeera came to prominence in the Western world following the September 11 2001 terrorist attacks in the US. The network was favoured by Arab terrorist organisations such as Al Qaeda for releasing footage of Western hostages and messages from leaders including Osama bin Laden.
As the invasion of Iraq began and US and British troops went into battle, the network angered the US and UK governments by broadcasting images of dead and captured coalition troops. Al Jazeera was then banned from operating its Iraq office by the interim Iraqi administration.
The head of Al Jazeera later had to step down after criticisms of its Gulf War coverage and allegations earlier that the Arab satellite television network was compromised by Iraqi spies.
But the network is moving towards a more reputable status in the West, with the launch of an English-language channel and hiring respected broadcaster Sir David Frost, who will join when it launches next spring.
However, it will face stiffer competition in future after the BBC World Service last month unveiled details of an Arabic-language TV station to rival Al Jazeera.
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