The controversy surrounds details of a leaked British government memo published by a UK newspaper, the Daily Mirror, earlier this week.
"This is a very serious charge with grave implications for the safety of media professionals," said Ann Cooper, executive director of the Committee to Protect Journalists.
"Refusing to address these reports in a substantive way only fuels suspicions."
The Mirror said Bush told UK Prime Minister Tony Blair at a White House summit on 16 April 2004 that he wanted to launch military action on Aljazeera's headquarters in Doha, Qatar.
Blair is reported to have talked the American president out of any such action, but the British government has refused to give any comment on the truth of the Mirror's story.
Instead British media have been warned that they face legal action under the Official Secrets Act if they report any further details from the memo.
The Mirror's report quoted an unnamed government official suggesting Bush's threat was a joke but added another unidentified source saying the US president was serious.
The White House has rejected
the report as 'outlandish'
Another media advocacy group, Reporters Without Borders said: "We find it hard to believe that President Bush really discussed this possibility. This would be extremely serious and would constitute a major and unprecedented violation of the right to information.
"If this report turns out to be true, it offers a new insight into the motives of the US forces, which have already bombed Aljazeera offices twice, in Afghanistan and Iraq."
Aljazeera staff in Doha and overseas bureaus planned to stage a symbolic protest over the report on Thursday.
"The staff of Aljazeera have decided to organise on Thursday a symbolic sit-in in front of the headquarters of the channel in Doha and its overseas bureaus to protest against this news," said Aljazeera journalist Youssef al-Shouli, who is also vice president of the Arab Association for the Defence of Journalists.
About 100 of the channel's journalists and employees have signed a petition calling on the broadcaster's board of governors to hold an official inquiry into the allegations.
They also demanded an immediate end "to attacks and incitement against Aljazeera and its employees" and called for "the opening of an inquiry into the bombing of Aljazeera's offices in Kabul and Baghdad".
There are fears that Blair-Bush
disputes may be made public
Aljazeera offices have twice been hit by US military action.
In April 2003, Aljazeera journalist Tariq Ayub died when the broadcaster's Baghdad bureau was struck during a US bombing campaign.
And in November 2001 Aljazeera's bureau in the Afghan capital, Kabul, was hit by a US missile.
On Wednesday, the British government's top lawyer, Attorney-General Lord Goldsmith, warned media organisations they would be breaking the law if they publish details of the leaked document.
Kevin Maguire, the Daily Mirror's associate editor, said government officials had given no indication of any legal problems with the story when contacted before publication.
Fresh revelations fear
"We were astonished, 24 hours later, to be threatened with the Official Secrets Act and to be requested to give various undertakings to avoid being injuncted," Maguire told BBC radio.
Aljazeera said in a statement that, if true, the Mirror's story would raise serious doubts about the US administration's version of previous incidents involving the station's journalists and offices.
"We demand an end to attacks and incitement against Aljazeera and its employees"
Vice president, Arab Association for the Defence of Journalists
The Guardian on Thursday reports that fears of fresh revelations about disputes between Bush and Blair on the Iraq conflict could damage the two nations' relations, prompted the unprecedented gagging of the media.
Senior MPs, Whitehall officials and lawyers were agreed that Lord Goldsmith had "read the riot act" to the media because of political embarrassment caused by a sensitive leak of face-to-face exchanges between the two leaders, reports The Guardian.
There were UK anxieties that US bombing in civilian areas in Falluja would unite Sunnis and Shias against British forces, adds the report.
Downing Street, the newspaper continued, stressed that the decision to take action was "entirely up to the attorney-general" and was intended to "draw a line in the sand" on further leaks.