Irish-born Hassan, who is married to an Iraqi, has been held by unidentified kidnappers since October 19 despite working in the country for years as director of CARE International.
The aid agency employed her through its Australian branch.
"I am terribly worried about her," Downer told commercial radio. "We still don't know who is holding her hostage."
On Wednesday militants threatened to hand the aid worker over to the group of Al Qaeda ally Abu Moussab Al Zarqawi within 48 hours unless British troops quit Iraq, Al Jazeera television reported.
The satellite network showed video footage of a hooded gunman and said the unknown group that had kidnapped Hassan had threatened to hand her over to Zarqawi's group within 48 hours "if the British government does not respond to its demands, chiefly the withdrawal of British forces from Iraq."
The presenter added that "Al Jazeera will refrain from airing the video for editorial and humanitarian reasons pertaining to the condition of the hostage."
Zarqawi, a fugitive Jordanian Islamist who has a $25-million US bounty on his head, has claimed responsibility for a string of deadly attacks in Iraq, including the beheading of foreign hostages.
In Dublin on Tuesday Irish Prime Minister Bertie Ahern told parliament that Al Jazeera had received a "disturbing" new video of Hassan.
"It is a distressing video which was decided by the station, on humanitarian grounds, not to show this. But they did make available early this morning the text of what was on the video," Ahern said.
Hassan, 59, the head of Iraq operations for leading charity CARE International, was snatched from her car on the way to work in the Iraqi capital on October 19.
She last appeared in a video aired on Al Jazeera last Wednesday appealing to London to withdraw its troops from Iraq and not to deploy around Baghdad.
She also asked for the release of Iraqi women prisoners.
No group has come out to claim responsibility for kidnapping Hassan.
© 2004 Agence France-Presse
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