Cease-fire talks to end the fighting in Gaza are stalled. Casualties in the enclave are mounting. And the families of hostages captured from Israel in the Hamas-led attack that ignited the war have grown increasingly desperate about the plight of their relatives in captivity.

After all the sticks Israel has wielded, it has decided to find out if carrots can make a difference.

Prime Minister Benjamin Netanyahu on Tuesday repeated his vow that Israel would hunt down and punish anyone who hurts a hostage, but he added a new promise: Israel will give a generous reward to anyone who returns a captive, paying $5 million and providing safe passage out of Gaza.

“I want to say to those who are holding our hostages: Whoever dares to harm our hostages will pay the price. We will pursue you and we will find you,” Mr. Netanyahu told Israeli troops on a visit to the Gaza Strip.

“To those who want to leave this entanglement I say: Whoever brings us a hostage, will find a safe way out for himself and his family,” he added. “We will also give $5 million for every hostage.”

The reward offer comes as members of Mr. Netanyahu’s office are facing scrutiny for allegedly leaking classified documents to influence public opinion, quell a popular push for a cease-fire that would win the hostages’ release, and promote the prime minister’s negotiating positions.