British newspaper The Daily Mirror reported that President Bush discussed bombing the Qatar headquarters of the popular Arabic network but that Mr Blair was opposed to the idea.
Both Mr Bush and Mr Blair hotly deny the claims.
But Britain has silenced all media outlets by warning any organization would be breaking the law if they publish details of the leaked document, said to show proof Mr Bush wanted to bomb the station.
Al Jazeera is critical of the US led wars in Afghanistan and Iraq.
The government’s top lawyer warned editors in a note after the Mirror newspaper reported a secret British government memo said British Prime Minister Tony Blair had talked Bush out of bombing the broadcaster in April last year.
Several British newspapers reported the attorney general’s note and repeated the Mirror’s allegations, which the White House said were “so outlandish” they did not merit a response.
Mr Blair’s office declined to comment.
Al Jazeera, which has repeatedly denied US accusations it sides with insurgents in Iraq, called on Britain and the United States to state quickly whether the report was accurate.
“If the report is correct then this would be both shocking and worrisome not only to Al Jazeera but to media organizations across the world,” the Arabic language station said in a statement.
The story would also be a shock for Qatar, a small Gulf state which cultivates good relations with Washington.
The Mirror said the memo came from Mr Blair’s Downing Street office and turned up in May last year at the local office of Tony Clarke, then a member of parliament for the town of Northampton.
Mr Clarke handed the document back to the government.
Leo O’Connor, who used to work for Mr Clarke, and civil servant David Keogh were charged on November 17 under Britain’s Official Secrets Act with making a “damaging disclosure of a document relating to international relations”.
The paper quoted an unnamed government official suggesting Mr Bush’s threat was a joke but added another unidentified source saying the U.S. president was serious.