Hillary Clinton, the US secretary of state, is on another mission to persuade Palestinians and Israelis to negotiate peace. But she may have to continue in the future without Mahmoud Abbas, the Palestinian leader, who twice last week reportedly said he was considering not running for a second term if and when new Palestinian elections are held.

With Washington apparently softening its position on an Israeli settlement construction freeze as a precondition for talks, Mr Abbas may feel he has been backed into a corner and can ill afford to back down.

His popularity has dived in recent weeks, first after reluctantly agreeing to meet the Israeli prime minister, Benjamin Netanyahu, in New York last month and then in the wake of the Goldstone report furore, when he reportedly bowed to US pressure to support a deferral of a UN vote.

Palestinians see continued settlement construction as a way for Israel to consolidate its occupation over parts of the West Bank and East Jerusalem and hence as contradictory to negotiations meant to secure an end to that occupation. Mr Abbas reportedly told Barack Obama, the US president, by phone last week that without a construction freeze he was inclined not to run for a second term. He is understood to have voiced the same sentiment in a meeting of Fatah’s Central Committee separately last week.

Analysts are divided over how serious Mr Abbas is. This is not the first time he has threatened to resign, and he is clearly trying to send a message to Washington that a man consistently hailed in the US as a “partner for peace” cannot always be asked to be the one to yield.

He has occasionally said that he never intended to have a second term in office. He is 74, and with the current bleak prospects for any political process with Israel he may feel he has reached a dead end. “I think that one has to take it seriously,” said Ghassan Khatib, head of the Palestinian Government Media Centre. “There are younger candidates in Fatah that the movement might see more appropriate for any new elections and Abbas has lost a lot of popularity recently.”

George Giacaman, a Palestinian analyst, noted that Mr Abbas had threatened to resign in the past for similar internal reasons and thought it unlikely that he was serious “at the moment”. However, “it is not impossible in the future that he will resign, especially if there is a failed political process, or one that hasn’t started.”

Mr Giacaman suggested Mr Abbas would continue for at least two years, the time frame Mr Obama laid out for securing a breakthrough in negotiations. But those negotiations have yet to get off the ground and waiting two years may not be an option for Mr Abbas, who is facing calls from within his own party for some kind of resistance over negotiations.

It is Mr Abbas’s staunch support for negotiations that has endeared him to Washington. A successor, not least Marwan Barghouti, the imprisoned West Bank Fatah leader, who consistently polls as the most popular Palestinian leader, will be unlikely to follow a similar negotiations-only strategy. That could undermine US attempts to get a peace process back on track in the near future.

“Who cares if Abbas resigns?” asked Yossi Alpher, an Israeli analyst. “Does Hamas care? I don’t think so. Does Netanyahu care? I doubt it. I don’t think he is interested in a serious peace process in any case. Do the Americans care? Well, they should.”