Some 200,000 people have fled intense fighting in Syria's second city Aleppo in the past two days, the UN has said.
UN humanitarian chief Baroness Valerie Amos said others were trapped in the city and needed urgent help.
Government forces launched a ground assault on Saturday after a week of sporadic shelling and sorties by fighter jets.
The BBC's Ian Pannell, in the Aleppo area, says residents are facing food shortages and power cuts.
He says the rebels are outgunned by the army, but they are fighting an effective guerrilla war in the streets.
Fighting has focused on the the Salah al-Din neighbourhood in Aleppo's south-west, where the rebels had embedded themselves.
Syrian state television showed footage from the city and interviewed soldiers who said they had taken complete control of Salah al-Din late on Sunday.
On Monday, officials in Damascus again said they had "purged" the area.
But activists have denied that the quarter has been overrun by the army, saying rebels are still in control.
They said fighting was continuing on Monday.
They also reported heavy shelling and clashes at the Sakhur quarter on the north-east side of the city centre, where another attack by government forces appeared to be under way.
Baroness Amos, speaking in New York, said that the Red Cross and the Syrian Arab Red Crescent estimated 200,000 people had fled the fighting in the past two days.
"It is not known how many people remain trapped in places where fighting continues today," she said.
"I call on all parties to the fighting to ensure that they do not target civilians and that they allow humanitarian organisations safe access."
She said many people had fled their homes to take shelter in schools and other public buildings.
Analysts say many others will have gone to nearby villages, and others will have fled across the border with Turkey.
The UN Security Council is chronically divided over Syria, with Russia blocking attempts by Western nations to ramp up pressure on Mr Assad.
France is due to take over the presidency of the Security Council this week, and Foreign Minister Laurent Fabius has pledged to continue pushing the issue.
He called Mr Assad an "executioner" and said he would ask for a ministerial level meeting of Security Council members before the end of the week.
US Defence Secretary Leon Panetta, who is on a five day Middle East tour, has also heavily criticised the government's assault on Aleppo.
He said the attack would be "a nail in the coffin" of President Bashar al-Assad.
Mr Panetta will visit Tunisia, Egypt, Israel and Jordan in a bid to reinforce the view that Mr Assad must step down.