The decision was made without holding a vote, with Finance Minister Bezalel Smotrich and other ministers being against the move.
Officially, security agency heads were not invited to attend given the meeting would be a briefing on PMs visits to Hungary and Washington.
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Israel also insists that, as part of such a deal, the mediators should be informed of the hostages' nutritional and health conditions.
The Israeli cabinet approved new roads in Ma'aleh Adumim to improve traffic and security, but Palestinians remain skeptical.
"Israel is certainly considering the possibility that these protests are a deception tactic by Hamas," an Israeli official said.
"This is a demand to the prime minister to stop sabotaging the agreement," the families said in an official statement.
Many Israeli officials, citing previous statements by Netanyahu, believe reaching the second phase of the deal would be more difficult to achieve.
Bar also said that the release of terrorists is expected to increase the motivation to carry out attacks in the West Bank.
Ministers from the Religious Zionists Party and Otzma Yehudit, as well as Likud Ministers David Amsalem and Amichai Chikli, voted against the deal.
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