Sergeant Major Ehud Goldwasser and First Sergeant Eldad Regev were abducted by militant group Hezbollah in July 2006 in a cross-border raid that sparked a month-long war in which more than 1,200 Lebanese civilians died.
Their bodies were returned to Israel on Wednesday in exchange for five living Lebanese prisoners and the remains of 199 Lebanese and Palestinian militants.
Haaretz said Goldwasser's funeral began in the morning in Nahariya, a Mediterranean resort town, and that Regev will be buried this afternoon at the military cemetery in Haifa.
The five released Lebanese militants received a heroes' welcome in their homeland on Wednesday, with thousands of Hezbollah supporters and political leaders gathering to meet them in southern Beirut.
The men include Samir Kantar, convicted of killing three Israelis including a four-year-old girl in 1979. Kantar, who was 16 at the time of the murder, had been behind bars for almost 30 years.
Hezbollah secretary general Hassan Nasrallah made a rare public appearance at the celebration in the Lebanese capital. Standing alongside Kantar, he called the release a "victory for the resistance."
"Our concern now is to free the rest of our lands... to protect our waters, our sovereignty and our honor," he said.
The prisoner exchange had been approved by the Israeli cabinet last week, with only three ministers voting against. At the time of the deal, it had remained unclear whether Goldwasser and Regev were still alive, and their deaths were not confirmed by Hezbollah until just before the handover.
Haaretz cited Medical Corps experts who examined the two bodies as saying the men had died during or shortly after their abduction. The men were fatally injured when Hezbollah fighters attacked their Hummer from close range with anti-tank missiles.
The unbalanced 'prisoner swap' has been harshly criticized by media and several politicians in Israel, over fears it will only encourage Hezbollah and Hamas to carry out further abductions, while giving them no incentive to keep their prisoners alive.