The council's 15 members are starting a detailed discussion of the resolution.
France and Russia - two of the council's five permanent members who have the power of veto - continue to have reservations about the wording.
They fear it could be used to authorise automatic military action if Iraq fails to co-operate with inspectors.
But the spotlight is now on the 10 non-permanent members of the council.
Syria is the only one flatly opposed to the resolution.
Diplomats say several others have sounded doubtful or critical of the United States, including Mexico, Guinea, Mauritius, Cameroon and Ireland.
Washington has had more positive signals from Bulgaria, Norway, Colombia and Singapore.
US hope
Iraq, which denies having weapons of mass destruction, says the US text amounts to a declaration of war.
"Its goal consists not only of attacking Iraq but also harming the prestige of the United Nations," said Iraqi Foreign Minister Naji Sabri.
US Secretary of State Colin Powell said on Thursday that he hoped to see America's new resolution - which is backed by the UK - voted in by early November.
Britain, America and China are the other three permanent members of the Security Council.
"I'm hopeful that... the United Nations will act in a powerful way and let Iraq know that it must obey or suffer consequences for continued disobedience," Mr Powell said at an international forum in Mexico.
Britain has warned that a vote on the draft might have to wait until after a briefing on Monday by the chief UN arms inspector, Hans Blix.
Objections
Russia immediately objected to the text released on Wednesday and claimed the backing of France and China.
UN Security Council
Five permanent members: US, UK, Russia, China and France
Current 10 rotating members: Mauritius, Mexico, Norway, Singapore, Syria, Bulgaria, Cameroon, Colombia, Guinea and Ireland
Ambassador Sergei Lavrov said Russia still objected to automatic threats and what he called an "unimplementable mandate" for the inspectors.
"Unfortunately, so far we have not seen changes in the text which would take into account these concerns," Mr Lavrov said.
French Foreign Minister Dominique de Villepin said in Brussels that the wording would need further attention.
"There is still work to do in New York and we need to do that before we can truly agree on a resolution," he said.
America's ambassador to the United Nations, John Negroponte, has predicted that Friday's talks will continue "all day if necessary".
Milder wording
The latest draft of the resolution is softer than a previous US text which authorised "all necessary means" to be used against Iraq if the work of the weapons inspectors was blocked.
The current version notes instead:
"The council has repeatedly warned Iraq that it will face serious consequences as a result of its continued violations of its obligations."
One apparent US concession in the new draft is that it no longer insists on troops from UN member-states accompanying weapons inspectors.