A Hezbollah military media unit said the Syrian army captured the town of Jisreen on Friday in an insurgent pocket in eastern Ghouta where thousands of people have fled since Thursday.
Lebanon's Prime Minister Saad al-Hariri has been invited to Saudi Arabia, his office said on Monday, setting the stage for his first visit since he abruptly resigned there in November.
Syrian state television showed a convoy of pro-government fighters entering the Kurdish-held Afrin region on Tuesday to help fend off a Turkish assault.
The leader of Lebanon's Iran-backed Hezbollah group, Sayyed Hassan Nasrallah, said on Friday the downing of an Israeli jet in Syria last week was a "very big military achievement".
Hezbollah's involvement in regional conflicts poses a threat to the security of Lebanon and is a destabilising influence in the Middle East, US Secretary of State Rex Tillerson said on Thursday.
The row has worsened a standoff between Berri, a Shia, and Aoun, a Maronite Christian, that threatens to ignite sectarian tensions in the run-up to a parliamentary election in May.
Israeli jets and ground-to-ground missiles struck Syria early on Tuesday, Syria's army said, as Israeli Prime Minister Benjamin Netanyahu reiterated he would do what was needed to stop Hezbollah gaining "game-changing" Iranian weapons.
Walid Jumblatt also condemned Iranian "dictates", an apparent response to a statement by the commander of the Iranian Revolutionary Guards this week that disarming of the Hezbollah was out of the question.
Lebanese President Michel Aoun said earlier this week that Hariri, who abruptly announced his resignation while in Saudi Arabia on November 4, was being held hostage by Riyadh.