He said the Iraqis has "stepped further back" from the group after the September 11, 2001, attacks in the US.
William Ehrman, the then director of international security at the foreign office, said the US had "put more weight on some of the links" than Britain did.
"But our view was there was no evidence to suggest serious collaboration of any sort between al-Qaeda and Saddam Hussein's regime," he said.
Dowse said the assessment was shared by colleagues in US intelligence, but suggested this may not have been the case among some in the Bush administration.
The second day of the inquiry, which is expected to last for up to a year, also addressed claims by the UK government that Baghdad was developing weapons of mass destruction in the lead up to the conflict in 2003.
Dowse said he had not been surprised by the 45-minute claim "because it didn't seem out of line [with assessments at the time]".
"It subsequently took on a rather iconic status which I don't think those of us who saw the initial report [expected] ... it wasn't surprising."
The 45-minute claim caused a major political row in Britain, after the BBC alleged that the intelligence dossier containing the claim was "sexed up" to strengthen the case for war.
The storm escalated when David Kelly, a government weapons expert, killed himself amid claims he was the source of the BBC story, prompting an official inquiry.
The five-panel inquiry, led by John Chilcot, a former civil servant, is investigating the justification for the war, how well military were equipped and trained and looking at lessons for future foreign policy.
Former senior officials from the foreign and defence ministries outlined on Tuesday Britain's policy towards Baghdad in early 2000.
The hearings are expected to climax with Tony Blair, the former British prime minister, taking the stand, at some point in the new year.