The measures are not as severe as U.S. sanctions in place and a proposed expansion of European Union steps against Damascus that would forbid the import of Syrian oil.
Diplomats said there are no plans for a vote yet.
Russia's U.N. Ambassador Vitaly Churkin has hinted that Moscow would use its veto power to knock down the draft if it was put to a vote at the present time. Western diplomats said that Russia and China were refusing to discuss the draft.
"The Russians say they have no instructions," a diplomat told Reuters on Friday on condition of anonymity.
As a result, Western diplomats in New York said their capitals would have to get involved to persuade Moscow and Beijing to join negotiations on the draft resolution to reach a consensus among the 15 Security Council members.
"Clearly we need this to be unlocked at the capital level because there is very strong resistance from Russia and China," a diplomat said.
Brazil, India and South Africa have also been reluctant to sanction -- or even condemn -- Syria, whose five-month crackdown on demonstrators has killed at 2,200, according to U.N. figures from earlier this week.
But one diplomat said that the three powerful developing countries, all three of which aspire to one day become permanent members of an expanded Security Council, were now "constructively engaging on the text."
The council had scheduled an informal closed-door meeting of ambassadors on the draft sanctions resolution on Thursday afternoon, but the Chinese and Russian envoys boycotted the meeting, diplomats said.
"You see two permanent members of the Security Council who deliberately leave an empty chair," a diplomat said. "We hope that they will engage constructively."
Earlier this week Russian Ambassador Churkin told reporters that it was not the time to sanction Syria. After months of helping block any council action on Syria, Russia and China had backed an August 3 statement by the Security Council that condemned the government's crackdown and called for it to end.
Russia, China, Brazil, India and South Africa have repeatedly complained that the NATO intervention in Libya has gone far beyond the U.N. mandate to protect civilians from violence by the government the council approved in March. They say they do not want the same thing to happen in Syria.
Russia has long had close ties to Syria and is one of its main arms suppliers. One of the proposed sanctions is an arms embargo that would make it illegal for Russian firms to sell weapons to Damascus.
The sanctions would impose a travel ban on 22 of Assad's family and associates and an asset freeze on 23 Syrians, including Assad. Envoys said President Assad was excluded from the travel ban in order to give him an escape route.
"We want him to leave the country," a diplomat said.
The United States and EU have urged Assad to step down.