The offices of the Hamas-operated Al-Aqsa television channel were also hit by the IDF following warnings.
On Monday, more than 300 rockets from Gaza were launched into southern Israel, the country's military said.
UN Secretary-General Antonio Guterres on Monday urged all sides to exercise maximum restraint.
There was no word from the security cabinet session whether Israel would agree, and rocket attacks continued amid Israeli air strikes as the forum met.
Israel and Hamas have traded tit-for-tat attacks regularly during the past few months, often when Hamas launches rockets in response to lethal Israeli army fire on weekly protests at the frontier - where Palestinians have been calling for an end to the blockade - or on Hamas positions after.
However, the recent tension since Sunday appears to be escalade since Israeli forces continue to deploy more forces to the border with Gaza.
Israel and militants in Gaza traded rocket and mortar fire Monday, a flare-up of violence prompted by an Israeli military operation over the weekend that resulted in the deaths of seven Palestinians and one Israeli.
Netanyahu has come under criticism from members of his right-wing government for his decision to allow Qatar to provide money to the Hamas-run Gaza Strip to pay for salaries and provide other support to ease a humanitarian crisis in the coastal enclave.
The heavy exchange of fire, which was triggered by a botched Israeli undercover raid into Hamas-ruled Gaza late Sunday, marked the most serious escalation since an Israel-Hamas war in 2014.
The escalation came hours after Palestinians and Israelis buried combatants who were killed on Sunday night, after what Israeli news media described as an Israeli intelligence mission inside the Gaza Strip that went awry.
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Islamic Jihad spokesman Daoud Shehab said the groups chose to retaliate for the deadly Israeli incursion "so the occupation and its supporters know that the lives of our sons come with a price". Israeli warplanes struck the Khan Younis area so the soldiers could retreat.
The raid and subsequent rockets follow several months of saber-rattling but relative calm between the two parties.
But by Monday, thick plumes of smoke were rising above Gaza and southern Israel.
One senior Israeli officer also was killed. It maintains a blockade on Gaza's coastline and has destroyed its airport, preventing any air transport in and out of the territory.
More than 170 demonstrators, most unarmed, have been killed by Israeli army fire in the confrontations in which some of the participants threw stones, burned tires or threw grenades toward Israeli forces.
Undercover Israeli special forces entered around two kilometres into the territory in a civilian auto near the southern Gazan city of Khan Younis, opening fire when they were discovered, the Al Qassam Brigades, Hamas's armed wing, said in a statement. Hamas responded by lowering the intensity of the border protest last Friday.
Prime Minister Benjamin Netanyahu raced back from First World War commemorations in France after Hamas hit back hard following a daring Israeli raid into Gaza that backfired.
Hamas is not only striking a new aggressive pose against Israel, it is also banking on a boost to its credibility in the Palestinian arena, through its willingness to clash head on with Israel, while it accuses the Palestinian Authority (PA), its hated domestic foe, of collaborating with Israel.
In a tweet after his arrival back home, Netanyahu praised the slain officer, whose identity was being kept confidential for security reasons, and said "our forces acted courageously".
Ismail Radwan, a Hamas official in the Gaza Strip, said that the Palestinian "resistance groups will halt all their activities once Israel stops its aggression".
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