Barake was a deputy commander of Hamas military wing, the Iz al-Din al-Qassam Brigades, Haaretz reported, citing Palestinian source.
Netanyahu was due to hold security consultations together with Liberman and senior officials from Israel's security agencies at the military's Tel Aviv headquarters Monday morning.
Israeli Prime Minister Benjamin Netanyahu has cut his trip to Paris for World War I commemorations short to return to Israel after the violence erupted along Israel's border with the Gaza Strip, a spokesperson said.
Their attempt follows months of Hamas-led riots on the Gaza border and arson balloons that have burned an estimated 10,000 acres of Israeli farmland, nature reserves and forests.
Palestinian militants on Monday fired dozens of rockets and mortar shells into southern Israel, critically wounding an Israeli teen in what appeared to be an act of revenge after a deadly undercover Israeli military raid the day before.
Israeli warplanes intervened by launching airstrikes over the area, immediately killing seven Palestinians and variously injuring more than seven others, according to the Palestinian Ministry of Health in Gaza.
IDF Chief of Staff Lt. Gen. Gadi Eisenkot called the mission "a very meaningful operation to Israel's security".
"Our fighters pursued the Israeli force and the security incident [clashes] is ongoing", read the statement.
The Israel-Gaza border fell quiet after a botched Israeli undercover operation in the Gaza Strip led to fighting that killed a Hamas commander, six other Palestinian militants and an Israeli colonel.
Hamas is demanding an expansion of the permissible fishing zone off the Israeli-controlled Gaza coast, uninterrupted flow of fuel to mitigate chronic power shortages and the lifting of Israeli restrictions on exports and imports.
Israeli critics of the transfer say it will benefit Hamas, with some dismissing it as extortion by the militant group, which is sworn to Israel's destruction.
Hamas has been leading protests since 30 March in a bid to ease an Israeli-Egyptian blockade imposed in 2007 in order to weaken the militant group.
"Two of them were intercepted by the iron dome", he said of Israeli air-defense system.
Rocket-warning sirens were later sounded in southern Israel.
At the weekend, Israel allowed the transfer of a reported $15 million in cash to the Gaza Strip from Qatar, partly for the payment of wages to government employees.
"Everyone knows where that money is going to go ultimately", said Netanyahu's agriculture minister, Uri Ariel.
Israel and Palestinian militants in Gaza have fought three wars since 2008, and recent months of unrest have raised fears of a fourth. More than 170 demonstrators, many unarmed, have been killed by Israeli army fire in the confrontations, in which numerous participants threw stones, burned tires or hurled grenades toward Israeli forces. IDF forces caught the infiltrator, who is being questioned by the Shin Bet (Israel Security Agency).