Rights activists said protests broke out after the main weekly Muslim prayers as the army pressed its campaign against northern towns and the number of refugees fleeing across the border into Turkey neared 10,000.
A senior US administration official put the toll at around 19 dead.
The United States is weighing whether war crimes charges can be brought against Syria to pressure its regime to an end to a bloody crackdown on dissent, the official said.
Other measures, including sanctions targeting the country's oil and gas sector, are being considered as part of a broader diplomatic campaign to increase pressure on the Syrian president.
US Secretary of State Hillary Clinton spoke by telephone Friday with Lavrov about a UN Security Council resolution on Syria, the State Department said, declining to comment further.
Russia joined China this week in boycotting UN Security Council talks on a draft resolution condemning Syria's crackdown on dissent despite mounting calls for stronger sanctions.
Rami Abdel Rahman, head of the Syrian Observatory of Human Rights, said five people were killed in the northern flashpoint city of Homs, two in the eastern city of Deir Ezzor and two in the Damascus suburb of Harasta.
Two other people were killed in Dael in the restive southern province of Daraa and one in the Damascus suburb of Douma, other activists told AFP by telephone. Several others were injured across the country.
According to Abdel Rahman "there was intense firing to disperse the demonstrations in Banias and there were casualties" but the head of the London-based Observatory was unable to give a breakdown.
About 5,000 protesters gathered in Homs, he said, adding demonstrations gripped several other cities and towns including Jableh in the west and in Suweida in the south, where club-wielding forces dispersed hundreds.
Protests also hit Latakia, Maaret al-Nooman and the countryside outside Damascus, activists said, as thousands took to the streets in the northwestern town of Qamishli, Amuda in the north.
Witnesses told AFP that a gunman opened fire on a police station in Rikn al-Deen, in Damascus, during a protest, killing a policeman and wounding at least four.
State news agency SANA also reported casualties among the ranks of the security forces.
"A member of the security forces was martyred and more than 30 were wounded by gunfire in Homs," the agency said.
It added two officers and four members of the security forces were wounded when gunmen attacked a recruitment centre in Deir Ezzor while three policemen were hit by gunfire in the Qabun neighbourhood of Damascus.
The military pressed ahead with its crackdown, sending tanks and troops into the northwestern town of Khan Sheikhun and surrounding villages, activists and witnesses said.
The Syrian army also attacked Janudiyeh, a few kilometres (miles) from the Turkish border, a Syrian activist helping the displaced people on the other side of the border told AFP by phone.
Nearly 10,000 Syrians have crossed the border into Turkey fleeing a crackdown by the Damascus regime, an official source said on Friday. About 1,200 arrived overnight, raising the number of refugees to 9,700.
The refugees are being settled in camps run by the Red Crescent in Turkey's southern Hatay province.
Some refugees began a hunger strike on Friday to protest against restrictions imposed by Turkish authorities, hours before Hollywood star and UN goodwill ambassador Angelina Jolie toured the camp, a Syrian rights group source said.
On the political front, France is seeking tougher EU sanctions against Assad's regime.
Discussions are underway with fellow EU states before a meeting on Monday of the bloc's foreign affairs committee, said foreign ministry spokesman Bernard Valero, adding that Syrian banks and private firms linked to regime figures could be hit.
The EU to date has slapped two sets of sanctions against Assad's regime, with EU foreign ministers in late May adding Assad to a blacklist of 23 Syrian officials hit by an assets freeze and travel ban.
The Syrian Observatory for Human Rights on Tuesday said the violence has claimed the lives of nearly 1,300 civilians and 340 security force members since it began in mid-March.
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