Moscow drew up the text yesterday endorsing the plan it spearheaded with the help of Turkey and Iran for a nationwide ceasefire.
The truce went into effect at midnight and appeared to be mainly holding Friday despite reports of sporadic clashes near Damascus.
The deal calls for subsequent negotiations in late January in the Kazakh capital Astana, Russia's UN ambassador Vitaly Churkin told reporters.
The council held closed-door consultations on the text yesterday.
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While Churkin voiced hope that the council could vote Saturday "and adopt it unanimously," diplomats said they were skeptical that such a vote would take place.
The ceasefire deal calls for negotiations over a political solution to end the conflict that has killed more than 310,000 people since 2011 and forced millions to flee.
The ceasefire -- which involves 13 groups representing 60,000 fighters who control "large chunks" of Syria -- appeared to be "holding adequately," the Russian envoy said.
The deal excludes jihadist groups including the Islamic State and Fateh al-Sham Front, an Al-Qaeda affiliate previously known as Al-Nusra Front.
Nevertheless, Moscow expects the UN will be "fully involved" in preparing for the Astana talks, he added.
"We hope others will join in, like Egypt, Saudi Arabia, Kuwait, Qatar," Churkin said.
Diplomats said they did not see how a quick weekend vote would occur.
"It needs to be studied seriously," said one Western diplomat.
There are still "a lot of unanswered questions," said another Western diplomat, adding that Russia might be hard-pressed to muster the nine votes needed for its resolution to pass.
It "stresses the importance of their full and immediate implementation and calls upon all parties to be guided by the aforementioned documents and provide support to their implementation."
As opposed to the first draft, the text also includes a reference to the talks being led by de Mistura, the UN special envoy.
It says the council views the eventual Astana talks as "an important part of the Syrian-led political process facilitated by the United Nations.