Secretary of State John Kerry began a round of intensive meetings here on Wednesday as he sought to forge a cease-fire to stop the bitter fighting in the Gaza Strip.

Mr. Kerry began his consultations by meeting in Jerusalem with Ban Ki-moon, the United Nations secretary general. He was scheduled to meet later in the West Bank with Mahmoud Abbas, the president of the Palestinian Authority, before heading to confer with Benjamin Netanyahu, the Israeli prime minister.

Mr. Kerry said at the start of his meeting with Mr. Ban that there had been some steps toward cementing a cease-fire but that “there is still work to do.”

The goal, Mr. Kerry stressed, was not only to obtain a cease-fire but to map out a “sustainable process going forward” that would address some of the long-term grievances of the Palestinians in Gaza and prevent the fighting between Hamas and Israel from erupting every few years. Mr. Kerry also remarked that he had arrived on the day of the funeral in Israel of Max Steinberg, an American who joined the Israeli military and who was killed in Gaza during the ground operation there.