The document, written by American secretary of state Colin Powell a year before the 2003 invasion, suggested the Prime Minister had signalled he "will be with us should military operations be necessary", the Press Association reported.
According to the Mail on Sunday, Powell wrote: "Blair continues to stand by you and the US as we move forward on the war on terrorism and on Iraq...
"Blair knows he may have to pay a political price for supporting us on Iraq, and wants to minimise it.
"Nonetheless he will stick with us on the big issues."
Former Tory Shadow Home Secretary David Davis told the Mail that the memos prove in "explicit terms what many of us have believed all along: Tony Blair effectively agreed to act as a frontman for American foreign policy in advance of any decision by the House of Commons or the British Cabinet".
Davis continued: "He was happy to launder George Bush’s policy on Iraq and sub-contract British foreign policy to another country without having the remotest ability to have any real influence over it. And in return for what?
"For George Bush pretending Blair was a player on the world stage to impress voters in the UK when the Americans didn’t even believe it themselves."
A spokesman for Blair told the newspaper the memo's content was consistent with what he had said publicly before.
The ex-premier told the Chilcot Inquiry that Bush would have understood he was ready to support military action if the diplomatic route was exhausted.
The briefing note, prepared for Mr Bush in March 2002 ahead of his Crawford summit with Blair later that year, emerged after a court ruling in the US led to the publication of thousands of emails received by Hillary Clinton.