The US has called on countries to decide where they stand on what it calls the Syrian regime's brutality.
Activists say 95 people were killed across Syria on Monday in cities including Damascus and Homs.
The White House said President Bashar al-Assad had lost control of Syria, adding "he will go".
Russia has said it will block a UN resolution calling for Mr Assad to hand power to a deputy who would then form a government of national unity.
Moscow said the text - proposed by the Arab League and backed by the US, the UK and France - was "not balanced" and would "leave open the possibility of intervention" in Syria's affairs.
The Local Co-ordination Committees, a network of anti-government groups, said on its Facebook page Monday's death toll was 95, including 72 in the central city of Homs. In Deraa, in the south, 15 people were killed, while six deaths were reported in the Damascus suburbs.
Their claims cannot be independently verified, as the the BBC and other international media are severely restricted inside Syria.
However, heavy machine-gun fire was reported in the restive Bab Amr district of Homs, while activists said at least 225 tank shells were fired at the eastern suburbs of Damascus.
Earlier, reports said the Syrian army had regained control of some Damascus suburbs recently held by rebel forces.
The White House said countries weighing their options at the Security Council should take into account that Mr Assad would be ousted.
"The regime has lost control of the country and will eventually fall," said spokesman Jay Carney.
Moscow, which has maintained its ties with Damascus, has so far resisted moves for a UN resolution condemning the violence in Syria. Russia has a naval base in the country and supplies arms to Syria.
"The current Western draft... certainly cannot be supported by us," Deputy Foreign Minister Gennady Gatilov told the Interfax news agency.
Mr Gatilov said the draft was "not balanced" and "leaves open the possibility of intervention in Syrian affairs".
France says 10 of the 15 countries on the Security Council now support the Arab League text. A minimum of nine council members must lend their backing in order for a resolution to be put to a vote.
However, Russia - as one of the five permanent council members - can veto any proposed resolution.
The BBC's Barbara Plett, at the UN, says Russia views the resolution as a first step towards regime change.
The UK has urged Moscow to reconsider its opposition.
"Russia can no longer explain blocking the UN and providing cover for the regime's brutal repression," said a spokeswoman for Prime Minister David Cameron.
On Monday, Russia also offered to mediate talks between the Syrian government and the opposition - a suggestion the opposition rejected out of hand.
The Syrian government has rejected the Arab League plan, which would see Mr Assad's deputy forming a national unity government within two months.
The prime minister of Qatar and the secretary-general of the Arab League are also going to New York to seek support for the draft text.
Qatar heads the League's committee dealing with the Syrian crisis and has previously called for Arab countries to send troops into Syria.
On Saturday, the Arab League announced it was suspending its month-old monitoring mission in Syria because of an upsurge of violence.
The Syrian army held funerals on Monday for 22 of its members killed in the previous 24 hours. The BBC's Jim Muir, in neighbouring Lebanon, says on average 20 members of the security forces are being killed each day.
Reports have emerged suggesting security forces may have killed senior army defector Lt-Col Hussein Harmoush, one of the first military officers to publicly declare his opposition to Mr Assad last year.
However, the Free Syrian Army, many of whose members are based in Turkey, said they could not confirm reports of his death.