Spokeswoman Libby Weiss said Navy Master Sgt. Shlomi Cohen was killed Sunday along the border where the second incident occurred, reported DEBKAfile, a website run by former members of Israeli military intelligence.
Weiss provided no details about the Lebanese soldiers' condition.
The Lebanese army had no information about Cohen's death, DEBKAfile said.
The Lebanese army was to begin its investigation into the sniper shooting in cooperation with Israeli and Lebanese liaison officers.
Arutz Sheva said the soldier who shot Cohen reportedly surrendered to Lebanese authorities voluntarily.
Citing Lebanese media, Arutz Sheva said the sniper disappeared following the attack then reported for duty at the Lebanese General Security post at the Ras al-Naqoura border crossing, where he surrendered.
The attacker is believed to have acted alone, Arutz Sheva said.
The Israeli military said it, too, was investigating the sniper shooting, saying it was prepared "for any action that will be required and will not tolerate any harm to Israeli citizens."