US Ambassador John Bolton said on Friday he expected the resolution to be put to a vote at a ministerial meeting of the Security Council on Monday.
Foreign ministers from 13 of the 15 council nations are expected to attend, Bolton said.
The draft resolution - co-sponsored by the United States, France and Britain - strongly backs a report by the UN investigating commission which implicated top Syrian and Lebanese security officials in the assassination of al-Hariri, and accused Syria of not cooperating fully with the inquiry.
The latest draft would require Syria to detain anyone the UN investigators consider a suspect, and allow the investigators to determine the location and conditions under which the individual would be questioned. It would freeze the assets and impose a travel ban on anyone identified as a suspect by the commission, "as a step to assist in the investigation" without prejudice to their guilt or innocence.
A committee comprising all Security Council members would be created to oversee the list of those subject to the travel ban and asset freeze, including approving exceptions.
Threat of sanctions
If Syria does not fully cooperate with the investigation, the draft says the council intends to consider "further measures", including sanctions, "to ensure compliance by Syria".
A UN inquiry into the murder of
al-Hariri has implicated Syria
While the text circulated late on Friday is in a final form that can be put to a vote, it can still be changed.
"What we see is unanimous support for a very strong, very clear signal, and we're just down now to a very few fine points," Bolton told reporters after council ambassadors went over a revised text at a closed-door meeting.
He said the resolution had the nine "yes" votes required for adoption, and would probably have more by the time of the vote. "I don't foresee a veto", he said.
Britain's UN Ambassador Emyr Jones Parry predicted the resolution will be "approved substantially".
Having foreign ministers adopt the resolution "is to show the intensity of the concern, and to make it very clear at the highest level what we expect", he said.
France's Jean-Marc de La Sabliere also said "there is a strong support for the text" and predicted approval.
But Russia and Algeria have opposed the threats of sanctions against Syria if Damascus fails to cooperate with the investigation.
"What we see is unanimous support for a very strong, very clear signal, and we're just down now to a very few fine points"
US ambassador to UN
"We have made it clear that it is premature and it is unjustified to talk about sanctions when the investigation is still going on and we do not know what will be the final outcome," Algeria's UN Ambassador Abdallah Baali said.
Russia's UN Ambassador Andrey Denisov said he was optimistic and hoped he would not have to use the veto, but reiterated Moscow's opposition to mentioning the threat of sanctions.
"We don't like it. We don't like it," he said. "We feel uncomfortable with that mention and our perception is that it should be deleted."
China has also said it opposes sanctions.
The latest text maintains the language on the sanctions threat, but it softens the wording on Syria's interference in Lebanese affairs, and changes the requirement that Syria make a commitment to stop supporting terrorism to a request.
Syria's UN Ambassador Fayssal Mekdad insisted Syria "has cooperated faithfully and sincerely" and will continue to do so.