WASHINGTON—While Israel was stepping up its attacks in Gaza on Monday, Israeli Prime Minister Benjamin Netanyahu raised the prospect of a "humanitarian cease-fire" in a phone call Monday night with U.S. Secretary of State John Kerry.

Mr. Kerry gave details of the conversation with Mr. Netanyahu at a news conference at the State Department Tuesday. Mr. Kerry added that Mr. Netanyahu has "consistently" said that he would embrace a cease-fire that allows Israel to protect against a network of cross-border tunnels...