The UN secretary general has condemned Syria for using tanks and live ammunition against protesters.
Calling for an independent investigation into recent killings, Ban Ki-moon said Syrian authorities had an obligation to respect human rights.
UN Security Council members have so far failed to agree a joint statement.
Reports from Syria suggest small anti-government protests are continuing around the country despite a government crackdown.
Hundreds of women have demonstrated in the southern city of Deraa, where the protests began six weeks ago, despite continued gunfire there.
They demanded the end of a government assault on Deraa.
In the coastal town of Baniyas the authorities did not try to stop a march in which participants expressed their solidarity with the people of Deraa.
Opposition activists say they are hoping for more demonstrations throughout the country after Friday prayers this week.
Around 400 people are said to have been killed over the last six weeks.
'Nothing to hide'
The 15-nation council is expected to take up the issue of Syria again later on Wednesday.
A draft text - proposed by the UK, France, Germany and Portugal - condemns the deadly violence against Syrian civilians and backs Mr Ban's call for a "transparent" independent investigation into deaths in the protests.
"It goes without saying that Syrian authorities have an obligation to protect civilians and respect international human rights. This includes the right to free expression and peaceful assembly," the UN chief said.
Mr Ban added that he was convinced that only an "inclusive dialogue and genuine reform" could address the Syrian people's legitimate aspirations, and restore peace and social order.
But speaking outside the Security Council chambers, the Syrian envoy to the UN said the country was perfectly capable of carrying out its own inquiry.
"Syria has a government, has a state," Bashar Ja'afari said, according to Reuters. "We can undertake any investigation by our own selves with full transparency... We have nothing to hide."
The BBC's Barbara Plett, at the UN headquarters in New York, says that though the UN's most senior human rights official has been invited to Damascus, it will be difficult for Security Council members to reach agreement.
Already, she says, there is great reluctance to support even a statement and veto-holders China and Russia are especially cautious as they are unhappy about intervention in Libya. Lebanon - closely tied to Syria - will also be reluctant to back a statement that condemns its neighbour.
Targeted sanctions
Earlier, European countries called for "strong measures" to halt repression in the country with France and Italy urging both the EU and UN to put pressure on Syria.
Despite the lifting of an emergency law last week, Syria's security forces have shot dead more than 400 civilians in their campaign to crush the month-long pro-democracy protests, according to Sawasiah, a Syrian human rights organisation.
It has called on the UN Security Council to start proceedings against Syrian officials in the International Criminal Court.
Western governments are struggling to find levers with which to influence Mr Assad, says BBC diplomatic correspondent Jonathan Marcus. The US already has wide-ranging sanctions against Syria and is now talking about additional targeted financial measures against named individuals.
European pressure may be more significant, adds our correspondent, as it is the Europeans who have been in the vanguard of bringing Syria in from the cold in both diplomatic and economic terms.
The BBC's Owen Bennett-Jones, in neighbouring Lebanon, says the Syrian government disputes the Western view that the demonstrations have been non-violent.
In a statement carried by the official news agency, it said it had sent troops to several cities on the request of citizens who were worried about "armed extremists".
