Secretary of State John Kerry, who has made 11 trips to Israel in the past year and has shuttled between seemingly endless late-night negotiating sessions with Israeli Prime Minister Benjamin Netanyahu and Palestinian President Mahmoud Abbas, acknowledged Thursday that the process was at a “critical moment” and might not survive. (Jacquelyn Martin/AP)

When his aides get discouraged about the prospects for Middle East peace, Secretary of State John F. Kerry often bucks them up with a phrase: “Don’t be afraid to be caught trying.”

But as his tireless efforts to broker Israeli-Palestinian negotiations hit bottom Thursday, with Israel’s cancellation of prisoner releases that were considered crucial to keeping the talks alive, there are some around Kerry — including on his senior staff and inside the White House — who believe the time is approaching for him to say, “Enough.”

Kerry risks being seen as trying too hard at the expense of a range of other pressing international issues, and perhaps even his reputation, according to several senior administration officials who spoke on the condition of anonymity about sensitive internal and diplomatic matters.

“A point will come where he has to go out and own the failure,” an official said. For now, the official said, Kerry needs to “lower the volume and see how things unfold.”

After negotiators met through the night, Israel announced that it would not release 26 long-serving Palestinian prisoners, the final members of a group of 104 whose freedom was part of last summer’s agreement to start talks. The Israelis said they were responding to a Palestinian decision to take unilateral steps to claim greater recognition as a state by the United Nations — a step that was itself a response to additional Israeli demands and an earlier delay in the prisoner release.

Kerry, who has made 11 trips to Israel in the past year and has shuttled between seemingly endless late-night negotiating sessions with Israeli Prime Minister Benjamin Netanyahu and Palestinian Authority President Mahmoud Abbas, acknowledged Thursday that the process was at a “critical moment” and might not survive.

“There’s an old saying: You can lead a horse to water, but you can’t make it drink,” he told reporters in Algiers before the Israeli announcement. “Now it’s time to drink, and the leaders need to know that.”

Amid the fast-moving events, all of them apparently downhill, some experts said it was not the time to walk away.

“First of all, you still have a clock that runs to the end of the month,” the original negotiating deadline set by Kerry, said Dennis Ross, who served as a senior Middle East expert in three U.S. administrations. “We’ve had a pretty significant investment in this, and to walk away from it before you determine for sure there is nothing else to be done, I’m not sure that’s what you want to do at this moment.”

“I have not been one of the skeptics” who have increasingly questioned the utility of the effort, Ross said. Although negotiations under previous administrations rarely approached the core controversies dividing Israel and the Palestinians, “I know from conversations with both sides that the kinds of things they’ve been talking about are the real issues,” including Israeli security, Jerusalem and the final borders between two states.

“In his defense, Kerry’s task is a thousand times greater than probably any previous negotiator,” said Aaron David Miller, a vice president at the Wilson Center who helped formulate U.S. policy in the Middle East for two decades before leaving the State Department in 2003.

“He’s not dealing with interim issues,” Miller said, but with the “crown jewels of the peace process.”

A U.S. official close to Kerry said the possible gains are worth any risk of failure or of looking overeager. Asked whether Kerry’s White House bosses might think he is obsessively chasing a lost cause, the official answered quickly: “That’s not what the president sees it as, and he’s the one who decides these things.”

Deputy national security adviser Benjamin J. Rhodes said, “The president was fully aware of Kerry’s interest and energy about the subject when he was chosen as secretary of state.” When President Obama went to Israel last spring, Rhodes said, “he went out of his way to both re-energize his own commitment to this at the beginning of his second term and also to very, very demonstrably empower Kerry.”

“Frankly,” Rhodes said, “having a secretary of state who plays such an active role . . . has, to some extent, taken the burden off of us.”

Obama charged during his 2008 campaign that George W. Bush had not devoted enough presidential time to Middle East peace. But the issue languished during Obama’s first term, and his first secretary of state, Hillary Rodham Clinton, was neither given nor claimed authority to promote negotiations.

Kerry is in a better position than Clinton ever was, Miller said. “He is risk-ready” and, unlike in Clinton’s case, “his political career is at an end.” But while Kerry’s efforts are “good and necessary,” Miller said, they are “not sufficient” to succeed if the parties themselves are unwilling. “The obituary will be that John Kerry cares more about this process than the Israelis and the Palestinians. I would argue that’s not a terribly positive legacy.”

Asked whether the secretary was trying too hard, Rhodes said, “There is no limit to the need to work with the Israelis and the Palestinians on this set of issues. . . . There is always going to be a reason to engage.”

“The broader question,” he said, “is how does this impact other things that we’re doing? If anything, Kerry has shown himself to be tireless, and still able to be fully invested in dealing with Ukraine, the Iran talks” and other issues. While there is an inevitable question of “whether he is able to do this and the other things on his plate,” Rhodes said, “thus far, in part because of his seemingly endless reservoir of energy and frequent-flier miles,” Kerry has pulled it off.

Yet however much Obama admires Kerry’s stamina and the worthiness of the goal, several officials said, the president believes the effort may be reaching its limit. Kerry, who discussed the issue with Obama last Friday when they met during a presidential visit to Saudi Arabia, still has the leeway to make his own decisions about what to say and do on the fly, officials said. But as the negotiations spiraled downward this week, he has spoken several times a day with Susan E. Rice, Obama’s national security adviser.

“If he goes too far, there’s the risk of looking desperate,” said the official who spoke of the need to “lower the volume” on the peace process. “He’s said many times internally that he’d like to resolve this situation, or at least get it on a stable road, so that he can go on to other things.”

Whatever the outcome for Kerry, the White House appears to see potential failure as having little downside for the administration overall. “Obviously, everybody would like it to succeed,” a second senior official said. But “people get that this is hard.”

Ross, who served three years in the Obama administration, was less certain. “It’s not like we have a lot of good things going on internationally right now,” he said.

“When other things aren’t going so well, this tends to look like just part of a piece. My own recommendation would be to think hard about how this affects everything else you’re doing.”

William Booth in Jerusalem contributed to this report.