Sir Christopher Meyer told the Iraq Inquiry that the two men spent an afternoon meeting in private at the former president's ranch in Crawford, Texas, in April 2002, which appeared to lead to a shift in the then Prime Minister's stance on Iraq.
Sir Christopher said: "I took no part in any of the discussions and there was a large chunk of that time when no adviser was there.
"The two men were alone in the ranch so I'm not entirely clear to this day what degree of convergence (on Iraq policy) was signed in blood, if you like, at the Crawford ranch.
"But there are clues in the speech Tony Blair gave the next day, which was the first time he had said in public ‘regime change'. He was trying to draw the lessons of 9/11 and apply them to the situation in Iraq which led - I think not inadvertently but deliberately - to a conflation of the threat posed by Osama bin Laden and Saddam Hussein.
"When I read that I thought ‘this represents a tightening of the UK/US alliance and a degree of convergence on the danger Saddam Hussein presented'."
Sir Christopher, who was Britain's ambassador to the US between 1997 and 2003, was called to give evidence about the changing nature of British and American policy towards Iraq in the two years before the invasion of March 2003.
Before the September 11 attacks on the US, Iraq was a low priority for the Bush administration, which was already "running out of steam", said Sir Christopher.
But the terrorist attacks immediately elevated Iraq towards the top of the US agenda.
"On 9/11 itself in the course of the day I had a telephone conversation with (then national security adviser) Condoleezza Rice and I said ‘who do you think did it?' She said: ‘There's no doubt it was an Al-Qaeda operation.' At the end of the conversation she said: ‘We're just looking at the possibility that there could be any link to Saddam Hussein.'
"That little reference to him, by the following weekend, turned into a big debate between Bush and his advisers."
Sir Christopher said hardliners in the Bush administration became increasingly convinced that Saddam was linked to Al-Qaeda, largely because of intelligence which proved to be wrong.
He said: "Paul Wolfowitz (then US deputy defence secretary) was quite convinced that there was a strong connection between Saddam Hussein and Al-Qaeda. There was a constant reference to the fact that Mohammed Atta (one of the 9/11 hijackers) had met Iraqi intelligence agents in Prague. That wasn't true, but you couldn't dig it out of the bloodstream of certain members of the US administration.
"There was another idea that there was an Al-Qaeda camp on the Iraqi border where Saddam would allow them to do things. That wasn't true either."
Sir Christopher said the US Department of Defense became so "irritated" by the CIA's "bias" against this incorrect intelligence that a "rival and replacement" in-house intelligence unit was set up by the White House.
The former ambassador said that when he first met President Bush in 1999, before he was elected, Mr Bush told him: "I don't know much about foreign policy. I'm going to have to learn pretty damn fast. I'm going to have to surround myself with good people."
Sir Christopher said Mr Bush's key advisers at the time - known as the Vulcans - included Condoleezza Rice and Paul Wolfowitz.
Mr Bush and Mr Blair immediately struck up a good relationship when they first met in 2001, said Sir Christopher, and during subsequent international conferences "Condoleezza Rice once said to me that the only human being (Bush) felt he could talk to was Tony, and the rest were creatures from outer space".
Sir Christopher said Tony Blair's speech immediately after 9/11, in which he promised to support America in its hour of need, "sealed Tony Blair's reputation in America, which remains sealed to this day".
"Wherever you went, people would rise to their feet and give you a warming round of applause. You had to be careful not to be swept away by this stuff."
Sir Christopher said Mr Blair's decision to support the US invasion was not "as poodle-ish" as has been suggested by critics, as he was "a true believer about the wickedness of Saddam Hussein" as early as 1998.
After Tony Blair came out in support of regime change in April 2002, Britain hoped Saddam could be removed by a combination of diplomatic pressure and the threat of force, which Mr Blair hoped would lead to Saddam either stepping down or being toppled by an internal coup.
But after President Bush set out a timetable for an invasion, the shortage of time meant that "instead of Saddam proving his innocence we had to prove his guilt by finding a smoking gun. We have never really recovered from that because there was no smoking gun".