DAMASCUS—Syrian government forces, backed by members of Lebanese militant group Hezbollah, launched a sweeping operation Sunday to capture a rebel stronghold near the Lebanese border, according to Syrian state media and activists opposed to the regime.

Taking the town of Qusayr, southwest of the city of Homs, would bolster recent gains by regime forces in central Syria and around the capital, Damascus. It also could further embolden Syrian President Bashar al-Assad, who told an Argentine newspaper over the weekend that his fate would be decided in elections scheduled for next year.

Mr. Assad's remarks complicate the renewed U.S.-Russian push for a political settlement to end the more-than-two-year civil war in Syria. Rebels and their Western and Arab backers insist any settlement must be predicated on Mr. Assad relinquishing power.

Underscoring the depth of the rift in Syria, regime supporters cheered the military operation in Qusayr, with some Assad loyalists openly calling for the government to storm rebel enclaves around the country, even if that means killing large numbers of civilians.

Mr. Assad's opponents accused the regime of carrying out a sectarian agenda in Qusayr.