The battles near the town of Quneitra in southwest Syria sent many residents fleeing, including dozens who crossed into neighboring Lebanon. The fighting in the sensitive area began Wednesday near the cease-fire line between Syrian and Israeli troops.
One of the worst-case scenarios for Syria's civil war is that it could draw in neighboring countries such as Israel or Lebanon.
There have already been clashes with Turkey, Syria's neighbor to the north. And Israel recently bombed targets inside Syria said to include a weapons convoy headed for Hezbollah in Lebanon, a key ally of the Damascus regime and an arch-foe of the Jewish state.
If the rebels take over the Quneitra region, it will bring radical Islamic militants to a front line with Israeli troops. The rebels are composed of dozens of groups, including the powerful al-Qaida-linked Jabhat al-Nusra, which the Obama administration labels a terrorist organization.
Israel has said its policy is not to get involved in the Syrian civil war, but it has retaliated for sporadic Syrian fire that spilled over into Israeli communities on the Golan Heights.
The Golan front has been mostly quiet since 1974, a year after Syria and Israel fought a war.
The Britain-based activist group Syrian Observatory for Human Rights said rebels seized control of parts of villages a few miles (kilometers) from the cease-fire line with Israel after fierce fighting with regime forces.
The Local Coordination Committees, another anti-regime activist group, reported heavy fighting in the nearby village of Sahm al-Golan and said rebels were attacking an army post.
The Observatory said seven people, including three children, were killed Wednesday by government shelling of villages in the area.
Rami Abdul-Rahman, who heads the Observatory, said the fighting around the town of Arnabeh intensified Thursday, a day after rebels captured it. He added that the rebels captured two nearby army posts.
In Lebanon, security officials said 150 people, mostly women and children, walked for six hours in rugged mountains covered with snow to reach safety in the Lebanese border town of Chebaa. They said eight wounded Syrians were brought on mules from Beit Jan and taken in ambulances to hospitals in Chebaa.
The officials, who spoke on condition of anonymity because they were not authorized to speak to the media, said the Syrians fled from the town of Beit Jan, near the Golan Heights.
The Yarmouk Martyrs Brigade, a rebel group active in southern Syria, said in a statement on its Facebook page that its fighters stormed an army post between the villages of Sahm al-Golan and Shajara.
Activists on Facebook pages affiliated with rebels in Quneitra announced the start of the operation to "break the siege on Quneitra and Damascus' western suburbs."
The fighting moved closer to Israel as President Barack Obama was visiting the Jewish state for the first time since taking office more than four years ago.
Associated Press writer Albert Aji in Damascus contributed to this report.