The pause in fighting enters a third day.

Halfway through a four-day pause in fighting that so far has brought two swaps of hostages and prisoners, Israeli leaders faced a dilemma over whether to restart the war against Hamas in the Gaza Strip once the truce ends on Tuesday morning.

Early on Sunday morning, Israel said it had received 13 Israeli hostages — eight children and five women — and four Thai citizens who had been held in Gaza, and had in turn released 39 Palestinians from Israeli prisons. The exchange followed an hourslong delay that raised fears the fragile deal could collapse.

The exchange was the latest step in an agreement that allows for a pause in fighting to be extended. Israel has said it is prepared to grant another day’s pause for every 10 hostages that Hamas releases beyond the 50 outlined in the agreement. Hamas has not responded to the offer.

“The question is Day 5,” said Alon Pinkas, an Israeli political commentator and former senior diplomat. “Does Israel resume the war?”

An extension that allows for more hostage releases could give further relief to Israelis who see the hostages’ freedom as the country’s biggest immediate priority. That sentiment could spread more widely among Israelis as each day of the cease-fire passes and more hostages are freed.

“And Hamas knows this very well,” said Shira Efron, a senior researcher at Israel Policy Forum, a New York-based political research group. “They’re going to play with Israel and say, ‘Oh, we found another five kids. If you give us another day, there are a few in the north that we can find.’”

But a longer pause could jeopardize the primary goal of Israel’s military campaign in Gaza: the destruction of Hamas, the armed group that led the Oct. 7 raid that killed an estimated 1,200 people in southern Israel and led to the abduction of roughly 240 hostages, according to Israeli authorities.

If the cease-fire lasts longer than four days, Hamas — which controlled all of Gaza until Israel invaded it last month — would have more time to regroup, allowing it to mount a fiercer defense when and if Israel renews its military campaign.

An extended cease-fire could also create more opportunities for other countries — particularly the United States — to pressure Israel to scale back its military goals. The Israeli response to the Oct. 7 attack has killed more than 12,000 Gazans, according to health officials there, leading to rising alarm among Israel’s allies about the conduct of its campaign.

President Biden, speaking on Friday in Nantucket, in Massachusetts, said “the chances are real” that the pause could open the door to a longer cease-fire.

But even if the United States pushes Israel to end or moderate its military campaign, Israeli leaders could simply ignore the criticism and plow ahead with the invasion.