United Nations (CNN) -- The four European members of the U.N. Security Council took the lead Wednesday in presenting a revamped resolution to the United Nations condemning Syria for its crackdown on peaceful anti-government demonstrators.
"It demands an immediate end to the violence and condemns diplomatic human rights abuses," said Mark Lyall Grant, Britain's ambassador to the United Nations. "It calls on the Syrian authorities to immediately lift the siege of affected towns, it calls for steps to address the legitimate aspirations of the Syrian people, which include reforms that will allow general political participation, inclusive dialogue and effective exercise of fundamental freedoms."
After the council met in private session, he told reporters that consultations would begin Thursday and expressed hope that it would be voted on within a few days. The resolution was introduced by Britain, France, Germany and Portugal.
France's ambassador, Gerard Araud, said rising violence in recent weeks has raised concern that the country could fall into civil war. The council "has to act," he said.
But gaining consensus may not be easy. U.S. Ambassador Susan Rice said several council members had raised the issue of Libya, where the effort by NATO forces to protect anti-government demonstrators appears caught in a stalemate. "Some things in this discussion, to be polite, strained credulity,and not to mention morality, and we will see where a number of council members come up," she said.
A recent French-British resolution failed to please Russia, a permanent member of the council, and some other members. In response, diplomats have tweaked the resolution language to make the measure more palatable to Russia and all members.
It was not clear precisely when a vote on the resolution would happen, though Britain and France said they would like it to take place by Friday.
Any resolution would fail with a veto by any of the five permanent members -- the United States, France, Britain, Russia or China.
In Washington, White House Press Secretary Jay Carney left no doubt about where the United States stands on the calls for reform by Syrian President Bashar al-Assad. "He can lead the transition, or he can get out of the way," Carney told reporters.
The Security Council has been criticized for failing to act to stem the violence in Syria as it did in Libya. But it wants to avoid pursuing military action as it did in Libya.
The new draft differs from the council's resolution on Libya, which endorsed the use of force by calling for "all necessary measures" to protect civilians under threat of attack by Moammar Gadhafi's government.
U.N. officials have expressed alarm over reports of torture, the use of live ammunition and shelling casualties across the country. Officials lately have focused on reports that children have been killed in the mayhem, which has resulted in more than 1,000 deaths.
Groups that include Amnesty International have urged the Security Council to refer the situation in Syria to the International Criminal Court. Syrian and international activists have visited the ICC at The Hague and supplied evidence of what they say are crimes against humanity by government forces.
CNN's Joe Vaccarello and Richard Roth contributed to this story.