TEL AVIV — A high-level Hamas delegation arrived in Cairo on Saturday as months of stop-start talks for a cease-fire and hostage release deal appear to have reached a crucial stage, even as Israel continued to threaten a ground offensive in Gaza’s southern city of Rafah that United Nations officials have warned will worsen the civilian death toll and deepen the enclave’s humanitarian crisis.

American and Egyptian negotiators indicated there had been signs of compromise in recent days.