BEIRUT, Lebanon — Deadly battles are raging in Homs, the central Syrian city that was long a hub and a symbol of the revolt against President Bashar al-Assad, with both soldiers and insurgents saying they have lost many fighters in recent days as the government presses an all-out assault on the last few rebel-held areas in the Old City of Homs.

Insurgents and antigovernment activists reported near-continuous bombardment of the Old City enclaves and the nearby neighborhood of Waer, a sometimes restive area whose population of 100,000 has been doubled by displaced Syrians, including many from the Old City, sheltering there.

State news media and government opponents reported that a car bomb killed numerous people on Friday outside a mosque in an area called March 8 Square near a government checkpoint. Each side blamed the other, giving death tolls ranging from none to more than 30. University students and Palestinian refugees living in the area were reported to be among the dead in the latest of many such bombings.

An army officer reached on the Old City’s outskirts said soldiers were advancing and taking casualties, but while officers had earlier predicted a quick victory, he declined to say when the fight would end. Government opponents and their international backers called on the United Nations to take action to prevent what they feared would be an imminent massacre in the Old City, saying that despite evacuations of about 1,500 civilians from the areas, more than 100 remained, along with more than 1,000 fighters. But with the Security Council divided, as Russia backs the government and the United States backs the opposition, only a tepid statement of “grave concern” was issued on Thursday.