The private note will remain secret - despite calls for it to be published by Iraq inquiry chairman Sir John Chilcot.
But Mr Blair, who is being grilled by the inquiry for a second time, summed up their contents for panel members.
He also revealed he disregarded Lord Goldsmith's warning that attacking Iraq would be illegal without further UN backing because it was "provisional".
The ex-PM said he believed his top legal officer would change his position on whether a second UN resolution justifying force was needed when he knew the full details of the negotiations.
'Difficulties'
Sir John said earlier this week the inquiry panel was "disappointed" the government would not allow the public release of statements Mr Blair made in July 2002 to Mr Bush and then US Defence Secretary Donald Rumsfeld.
The former prime minister said that although he agreed with cabinet secretary Gus O'Donnell, who blocked their release on the grounds that they would compromise diplomatic confidentiality, he was "not going to hide behind the cabinet secretary".
Summing up the contents of the statements, he said he had told Mr Bush: "You can count on us, we are going to be with you in tackling this, but here are the difficulties."
The message he wanted to get across, he added, was "whatever the political heat if I think this is the right thing to do, I am going to be with you, I am not going to back out if the going gets tough. On the other hand, here are the difficulties and the UN route is the right way to go".
He said his foreign policy adviser Sir David Manning, who earlier told the panel he thought the statement was too sweeping and wanted the wording changed, would have preferred him not to have given "undertakings" to President Bush.
Mr Blair was also quizzed about apparent discrepancies between what he told the committee in January 2010 and recent statements to the committee by his Attorney General Lord Goldsmith.
'Uncomfortable'
Lord Goldsmith said he had been "uncomfortable" with statements Mr Blair made in the Commons ahead of the war suggesting Iraq could be attacked without UN authorisation, when he was warning at the time that such a move would be illegal.
Mr Blair told the panel he was also "uncomfortable" at the time, as he was trying to make the "political" case for taking action against Saddam.
"In the end I wasn't making a legal declaration, but a political point - if there was another breach we had to act," he told the panel.
He said "I was trying to hold that line in circumstances where it was very difficult" and if UK legal disagreements had emerged it would have wrecked ongoing negotiations.
"If a chink of light had opened up it would have been a political catastrophe for us," said Mr Blair.
But he said he was convinced that if Lord Goldsmith spoke to Sir Jeremy Greenstock, the UK's then ambassador to the UN, and spoke to "the Americans" he would change his mind, which turned out to be the case.
Mr Blair issued a 26 page written statement ahead of his appearance in response to more than 100 detailed questions from the inquiry panel, in which he set out the process by which he said Lord Goldsmith changed his mind.
On 14 January, 2003, Lord Goldsmith told Mr Blair in a six-page draft legal opinion that UN Security Council resolution 1441, which declared Iraq in "material breach" of its obligations to disarm, was not enough on its own to justify the use of force against Iraq.
'Material breach'
Mr Blair said in the statement to the inquiry: "I had not yet got to the stage of a formal request for advice and neither had he got to the point of formally giving it.
"So I was continuing to hold to the position that another resolution was not necessary."
The former prime minister said he was aware of Lord Goldsmith's concerns about the legality of attacking Iraq.
But he added: "I believed that he would, once he was abreast of the British but most of all the US negotiating history, conclude that 1441 meant what it said: Saddam had a final opportunity to comply, failure to do so was a material breach, and that revived the earlier resolutions authorising force."
Mr Blair was asked about a memo, dated 30 January 2003, in which Lord Goldsmith repeated that resolution 1441 did not authorise the use of military force against Iraq on its own.
A scribbled note next to this sentence in the former prime minister's handwriting says: "I just don't understand this."
Referring to this comment, Mr Blair said in his statement: "I did not understand how he could reach the conclusion that a further decision was required when expressly we had refused such language in 1441."
'Gung ho'
The inquiry also released a note from Mr Blair to Jonathan Powell, his chief of staff, shortly before his visit to then US President George Bush's ranch in Crawford, Texas, in April 2002, in which he argued that Labour should be "gung-ho" about dealing with Saddam Hussein.
In the note, Mr Blair said that, from "a centre-left perspective", the case for action against the Iraqi dictator should be "obvious".
"Saddam's regime is a brutal, oppressive military dictatorship. He kills his opponents, has wrecked his country's economy and is a source of instability and danger in the region," he wrote.
"I can understand a right-wing Tory opposed to 'nation-building' being opposed to it on grounds it hasn't any direct bearing on our national interest.
"But in fact a political philosophy that does care about other nations - eg Kosovo, Afghanistan, Sierra Leone - and is prepared to change regimes on the merits, should be gung-ho on Saddam."
Mr Blair said the meeting with Mr Bush at Crawford "did not result in an alteration of policy".
The evidence session is to last more than four hours.
Mr Blair's previous appearance prompted demonstrations at Westminster, although the former prime minister arrived hours before the start and avoided any confrontation.
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