The UNs urged all sides to step "back from the brink" after months of increasing tensions, with a fourth Palestinian shot dead during protests along the border with Israel. The soldier was the first Israeli killed in that time.
Israel says it is defending its sovereign border and accuses Hamas of using the protests as cover for attempts to breach the border fence and attack Israeli civilians and soldiers.
Almost 150 Palestinians have been killed since the demonstrations began on 30 March, a lot of them shot by Israeli snipers.
The flare-up is the latest as demonstrations and clashes on the frontier since March have seen at least 149 Palestinians killed.
The unnamed official claimed "Hamas suffered a serious blow yesterday, and requested a ceasefire via Egypt, while promising to stop the arson terror and terror at the border fence".
Israel has not officially confirmed the cessation of hostilities.
The Palestinian group says through Egyptian and United Nations efforts it has been agreed to "return to the era of calm".
But the Hamas source claimed incendiary balloons and kites, which Palestinians have been floating over the border for months to spark fires inside Israel, were not included in the agreement.
Hamas has said a deal to restore calm in the besieged Gaza Strip has been reached, ending the Israeli onslaught on Hamas positions and other violence that resulted in the deaths of four Palestinians and one Israeli soldier on Friday.
Overnight Saturday, Hamas and Israel agreed to a ceasefire after five people including an Israeli soldier were killed Friday.
But more recently, some Gaza officials said the protests would end if Israel lifted an economic blockade of the Gaza Strip.
Israel said its aircraft and tanks had struck dozens of targets belonging to Hamas and across the Gaza Strip, including a drone warehouse, aerial defence systems and observation posts.
Palestinians are calling to return to the homes their families were forced from in 1948, during the war surrounding the creation of Israel. The military wing of Hamas, which has fought three wars with Israel since 2009, said three of its fighters were killed in the strikes.
Prime Minister Benjamin Netanyahu of Israel and his defense minister, Avigdor Lieberman, arrived at military headquarters in Tel Aviv on Friday night for emergency meetings with top officials, a highly unusual event for the start of Sabbath.
The ceasefire is the second since in a week. He said they would be returned only as part of a deal similar to that in 2011 under which captured Israeli soldier Gilad Shalit was released in exchange for more than 1,000 Palestinians jailed in Israel.
Israeli politicians have been calling for a fierce response to the kites and balloons, which have caused damage amounting to millions of shekels (hundreds of thousands of dollars).
Khalil al-Hayya, Hamas's deputy chief in Gaza, said the kites would continue to fly.More news: Twitter chief Jack Dorsey caught up in purge of fake followers