Former British Cabinet Office civil servant David Keogh, 49, and Leo O'Connor, a 42-year-old former parliamentary researcher, are both accused of breaching Britain's Official Secrets Act.
A judge at Bow Street Magistrates Court, central London, sent the case to the capital's Central Criminal Court at the Old Bailey where a preliminary hearing is scheduled for January 24.
The memo, details of which were published in British newspapers last November, allegedly recounted how Tony Blair, the British prime minister, dissuaded Bush from bombing Aljazeera's Qatar headquarters.
The White House, which has been angered by Aljazeera's reporting of the Iraq war, dismissed the report as "outlandish" but it prompted protests in a number of Arab countries.
Speaking outside court on Tuesday, O'Connor's lawyer, Neil Clark, said defence lawyers had been shown the document at the centre of the case for the first time. It was four pages long and marked "secret", he added.
Clark said he had signed a guarantee not to discuss the document with anyone other than his client.
"We will be seeking disclosure of the document ... and that will depend on the (trial) judge," he said.
Aljazeera's Iraq reporting had
angered the White House
The document was "crucial" to the case, he went on, adding: "I did not think there was anything in there that could embarrass the British government."
Mark Stephens, a lawyer observing proceedings for Aljazeera, said: "Hopefully, (the document) will be revealed for all to see in court in due course so people can make an assessment of the real risk of what took place.
"Obviously, if there was a suggestion that journalists be killed that would be counselling or procuring that to happen.
"Many (British) expatriates, former members of the BBC and CNN are working in Doha and would have been killed if this was carried out.
"They (Aljazeera) are obviously concerned for the safety and well-being of their staff and for the children that visit the Aljazeera's children's channel every day."