The Security Council statement, initially drafted by France in mid-November and now transformed into a resolution, is finally expected to come to a vote on Tuesday after the US agreed to a compromise.
Based in The Hague in the Netherlands, the court is the first permanent world tribunal set up to prosecute individuals for war crimes, genocide and other gross human rights abuses.
It came into being last year and 97 countries have ratified the 1998 statute creating the tribunal. Of the 25 European Union nations, only the Czech Republic has not submitted its ratification papers.
Washington, which fears politically driven prosecutions of its officials serving overseas, calls the court "fatally flawed" and has been campaigning hard in recent weeks to prevent it from becoming a routine part of UN operations.
US diplomats campaigned unsuccessfully to have the tribunal taken off the agenda of the UN General Assembly and have fought to prevent the use of UN funds to support it, even trying to bar discussions of it in the world body's meeting rooms.
French ambassador Jean-Marc de la Sabliere complained about the US drive during a 15 November closed-door council meeting after the US blocked his draft statement on the investigation into the Burundi massacre, council diplomats said.
Burundi's government has been trying for more than three months to fix blame for the 13 August slaughter of more than 160 ethnic Tutsi Congolese in the Gatumba refugee camp.
Washington argued the language of the French draft's offer of "international support as appropriate" was a hidden reference to the International Criminal Court and blocked its approval.
In the compromise reached on Monday, the council linked the possibility of international support to a commitment by the Burundian government to quickly wrap up its investigation.