GAZA CITY, Gaza Strip ? Palestinians and Israelis moved closer to halting more than four years of bloody violence on Sunday, with the new Palestinian leader saying he had made progress toward a truce deal with militants. Israel pledged for the first time to hold fire if calm prevails.

Palestinian leader Mahmoud Abbas extended his stay in Gaza until today, trying to win a declaration from armed groups that they will halt attacks against Israelis. Abbas has been meeting representatives of Hamas, Islamic Jihad and the Al Aqsa Martyrs’ Brigades, a group with ties to his Fatah movement.

Israeli leaders said Sunday they would suspend their military operations if the Palestinians maintained calm, taking a major step in the direction of a cease-fire.

As in the past, the Palestinians were negotiating among themselves about conditions for halting violence without directly involving Israel — but making parallel demands on the Israelis.

A truce was not formally in place, but little violence was reported Sunday, as about 3,000 Palestinian police patrolled Gaza areas near the Israeli border for a third day to prevent militants from firing rockets.

Though there have been other brief periods of quiet during four years of violence, optimism was heightened this time. Abbas, who succeeded Yasser Arafat after winning a Jan. 9 election, was working around the clock to stop violence, backing up his earlier statements that violent Palestinian resistance had been a mistake.