Mr Bremer released a series of letters to the New York Times detailing the need to take "robust" action against the toppled regime of Saddam Hussein.
But in a book published on Tuesday, Mr Bush says the agreed policy was to keep Iraq's army intact.
The disbanding is now widely seen as a mistake that fuelled Iraq's insurgency.
In a letter dated 22 May 2003, Mr Bremer, the former head of the US-led civilian administration in Iraq, appears to propose the dismantling of the army to Mr Bush.
Letter to Paul Bremer, May 2003
"I will parallel this step [dismantling the Baath Party] with an even more robust measure dissolving Saddam's military and intelligence structures to emphasise that we mean business.
"We are seeing signs that the outlawed organisations are behind some of the street violence here."
The next day Mr Bush replies, according to a second letter published in the Times: "Your leadership is apparent. You have quickly made a positive and significant impact. You have my full support and confidence."