After Hamas’s political leader, Ismail Haniyeh, was assassinated in late July, Hamas officials told Qatari mediators that they had new demands for the already deadlocked cease-fire talks, according to one Arab and one American official.

The suggestion worried the Qatari prime minister, who had spent months urging Hamas to compromise. With the support of his staff, he pushed back in meetings and calls with the Palestinian militant group, the officials said.

Hamas ultimately dropped the idea.

As the talks for a cease-fire and the release of hostages have stalled and sputtered in recent months, Qatar has leveraged its influence with Hamas in an effort to break through myriad impasses, according to interviews with more than a dozen officials with knowledge of the negotiations, including ones from the region and from the United States. Most of the officials spoke on condition of anonymity so they could share details of the closed-door discussions.

Since the war began, Qatar, along with Egypt, has emerged as a crucial mediator between Israel and Hamas, hosting marathon discussions with Palestinian representatives in air-conditioned rooms in downtown Doha, the Qatari capital, and channeling messages from the Biden administration to Hamas. The Qataris have also been working with the Israelis, even though the countries do not have formal diplomatic relations.

The Qatari efforts have taken on more urgency as the negotiations appear at a standstill. Hamas and Israel remain far apart on a deal — and the goal posts seem to be constantly shifting.

Two American officials said Hamas in recent days had added new demands for the release of hostages, asking for more Palestinian prisoners to be released in the opening phase of the agreement.