Russia has called for ?regular and obligatory? breaks in the air strikes
The council?s president, Dina Kawar, who is also Jordan?s UN ambassador, said members needed time to ?reflect on the proposal?.
The air strikes against Shia Houthi rebels continued for an 11th night on Saturday.
The Red Cross has called for a 24-hour ceasefire to bring in desperately needed medical supplies.
It says medical teams and rescuers must have access to Aden, a stronghold of those loyal to President Abdrabbuh Mansour Hadi, or more civilians will die.
Countries including Algeria and Pakistan have stepped up the evacuation of their citizens from Yemen, as the humanitarian situation deteriorates.
A Pakistani aircraft will fly to the capital Sanaa on Sunday to pick up 170 people. More than 800 Pakistani citizens have already left. Flights from China, Egypt, Sudan and Djibouti are also scheduled for Sunday.
Algeria rescued 160 of its citizens along with other north Africans on Saturday.
?Strewn with dead bodies?
The Houthis, who took control of Sanaa in September, have made significant gains in Aden, forcing President Hadi to flee to Saudi Arabia.
In the past few days rebels pushed through to the heart of the port city using tanks and armoured vehicles. They were repelled from the presidential palace by loyalists supported by Saudi air strikes and weapon drops.
The Red Cross has reported a dire situation, with the streets of the city ?strewn with dead bodies? and the wounded ?streaming? into hospitals and clinics.
At least 185 dead and 1,282 wounded have been counted in hospitals in Aden since 26 March, health department director Al-Kheder Lassouar said.
The toll does not include rebel casualties or victims of air raids, he added.
On Friday, Russia presented the draft resolution to an emergency session of the UN Security Council in New York. It calls for ?regular and obligatory? breaks in the bombings to allow the rescue of foreigners caught up in the fighting.
The draft, however, makes no mention of a halt to fighting by the Houthis.
It also demands ?rapid, safe and unhindered humanitarian access to ensure that humanitarian assistance reaches people in need?.
Meetings on Saturday led to nothing concrete, but Ms Kawar said she ?[hoped] that by Monday we can come up with something?.
?More will die?
Over the past two weeks, fighting in Yemen has left more than 500 people dead and about 1,700 wounded, UN humanitarian chief Valerie Amos said.
Robert Mardini, head of operations for the International Committee of the Red Cross in the Middle East, said: ?Our relief supplies and surgical personnel must be allowed to enter the country and safely reach the worst-affected places to provide help.?
?Otherwise, put starkly, many more people will die.
?For the wounded, their chances of survival depend on action within hours, not days,? he added.
The Red Cross says it has 48 tonnes of medical equipment to treat up to 3,000 people, ready to leave pending clearance.
Britain?s deputy UN ambassador Peter Wilson expressed regret for civilian casualties but said Britain would carry on supporting Saudi-led military action against the Houthi rebels.
The Houthis are also supported by troops loyal to the former president Ali Abdullah Saleh, who was ousted in the Arab Spring protests.
Meanwhile, armed tribesmen pledging to restore security have entered the south-eastern city of Mukalla taken over by al-Qaeda militants at the end of last week, Reuters news agency reported.