Nearly 60 countries led by Switzerland have expressed support for a proposal to refer the conflict in Syria to the International Criminal Court (ICC).
They are calling on the ICC to investigate possible crimes against humanity and war crimes.
Switzerland's permanent representative to the UN Paul Seger issued an appeal to the Security Council to adopt a draft resolution over the issue.
However, such a resolution is likely to be opposed by China and Russia.
Syria is not a party to the Rome Statute, the treaty establishing the ICC.
France has been arguing that the ICC should investigate crimes against humanity in Syria.
Correspondents say that if Beijing and Moscow veto the proposal, it will be the fourth time they have fended off Western resolutions at the UN since the Syria crisis erupted three years ago.
Last month graphic photographs of prisoners allegedly tortured by Syrian government forces were shown to members of the UN Security Council.
The images - reportedly smuggled out by a defector and dismissed by Damascus - appeared to show evidence of abuse, including beatings, strangulation and long-term starvation.
"The United Nations - the Security Council and the wider membership alike - should be united in acting for justice and accountability for the victims of the conflict in Syria," Mr Seger wrote in the letter.
Diplomats say that while the US did not sign the letter, it supports the French proposal.
Western powers say that the list of crimes committed by Syrian government forces include systematic torture of civilians, chemical attacks and the use of "barrel bombs" packed with explosives.
To truly end this suffering, it will take the agreement of all warring parties to a conflict that has exacted an unbearably heavy human cost
Jihadist rebels have meanwhile been accused of carrying out mass executions of detainees.
More than 150,000 people have been killed since the Syrian conflict began, with millions forced to flee their homes.
France wants the UN vote to take place on Thursday, but Russia has already said it will oppose the resolution.
China joined Russia in vetoing the three previous Syria resolutions.