Witnesses said at least three people were shot dead at a funeral north of the city of Homs. Rallies were reported in Aleppo, Baniyas, Deraa and nearby Suwaida.

Some 200 people have been killed and dozens arrested in weeks of protests.

On Saturday, Mr Assad said he expected the country's 48-year-old emergency law to be lifted by next week.

The BBC's Owen Bennett-Jones in neighbouring Lebanon says the reaction so far suggests that at least some people are not yet satisfied with the extent of the concessions made by the government.

Witnesses said demonstrators in Deraa gathered in a main square after noon prayers and were heard calling for "the downfall of the regime", according to AP news agency.

The rallies were held on Syria's Independence Day, which marks the anniversary of the departure of French soldiers 65 years ago.

In the Talbiseh district, north of the central city of Homs, security forces opened fire on a funeral procession, killing at least three people and injuring dozens of others, eyewitnesses said.

The funeral was for a man killed by security forces the day before, witnesses told Reuters.

Meanwhile, a woman at the demonstration in Suweida said they had chanted "God, Syria, freedom, that's all", but then were confronted by supporters of President Assad.

"They came at us with sticks and also hit us with the pictures they were carrying of Bashar - the same president who was talking about freedom yesterday," she told AP.

In Hirak, 33 km (20 miles) north-east of Deraa, angry Syrian mourners chanted slogans against Mr Assad at the funeral of 20-year-old soldier Mohammad Ali Radwan al-Qoman.

A relative of Qoman said the authorities had told them he had been accidentally electrocuted at his military unit, but that they had seen signs of beatings and believed he had been killed by security forces.

Demonstrations were also reported in the port city of Baniyas and Syria's second city, Aleppo.

On Saturday, Mr Assad told the cabinet a legal commission asked to examine the lifting of the emergency law had come to its conclusions.

"I think the commission has finished its work, on Thursday, and the recommendations will be given to the government so that they become law immediately. I don't know how many days it will take you and I think that the maximum deadline for the lifting of the state of emergency will be next week," he said.

New security legislation would be introduced in place of the law, he said, adding that the new government should also study ideas for a multi-party system and greater press freedom.

The lifting of the emergency law - which bans public gatherings of more than five people - was a key demand of the protesters.

Mr Assad's speech followed a massive demonstration on Friday after prayers, when tens of thousands of demonstrators rallied in the capital, Damascus.

Mr Assad blames the violence in recent weeks on armed gangs rather than reform-seekers and has vowed to put down further unrest.