In one incident, more than 30 troops were killed in clashes with suspected army defectors in a southern town near the Jordan border, activists say.
King Abdullah became the first Arab leader to openly urge Mr Assad to quit.
He told the BBC that if he were in Mr Assad’s position, he would start talks to ensure an orderly transition.
“I would step down and make sure whoever comes behind me has the ability to change the status quo that we’re seeing,” King Abdullah stated in an exclusive interview with BBC World News television.
He said: “If Bashar [al-Assad] has the interest of his country [at heart] he would step down, but he would also create an ability to reach out and start a new phase of Syrian political life.
Angered by the king’s comments, about 100 Bashar supporters rallied outside the Jordanian embassy in Damascus late on Monday.
Three protesters scaled the embassy fence and took down the Jordanian flag, Jordanian Foreign Ministry spokesman Mohammad Kayed was quoted as saying by the Associated Press.
The spokesman added that no-one injured during the incident.
Many Arab leaders have condemned the crackdown on months of protests in Syria, and the Arab League voted on Saturday to suspend Syria’s membership.
Syrian Foreign Minister Walid al-Muallem has reacted defiantly to the Arab League’s move, saying it was illegal and vowed to overcome “conspiracies” against Damascus.
The UN says more than 3,500 people have died since the start of the protests in March. The Syrian authorities blame the violence on armed gangs and militants.
In the latest violence, 27 civilians were shot dead by security forces in the flashpoint southern province of Deraa on Monday, the UK-based Syrian Observatory for Human Rights said.
In the same area, 34 government soldiers died in clashes with suspected army defectors, the Observatory said. It added that 12 deserters were also killed.
Another four civilians were killed in the central city of Homs, activists said.
Such claims are impossible to verify as the Syrian government has severely restricted access for foreign journalists.
Many Western powers have urged President Assad to stand down. Both the EU and the US have said he has lost legitimacy but have ruled out military intervention.
It added 18 Syrian officials to a list of people affected by a travel ban and asset freeze. This brings to 74 the number of President Assad’s inner circle who have been blacklisted.
The EU also approved the freezing of loans to Syria from the European Investment Bank.
French Foreign Minister Alain Juppe said he hoped the UN would finally impose its own sanctions on Syria.
Russia and China last month vetoed a Western-sponsored UN Security Council resolution condemning Damascus.
The US welcomed the moves by the Arab League and the EU.
The Arab League is set to hold another meeting to discuss Syria on Wednesday.
The league has already proposed sending a mission of 500 human rights and military observers, and also journalists to monitor the situation on the ground in Syria.