U.S. Secretary of State John Kerry urged Hamas on Tuesday to pursue a negotiated end to its conflict with Israel after what he said were constructive talks with Egyptian officials about their cease-fire proposal.

After meeting Egyptian President Abdel Fattah el-Sisi, whose country has some leverage over Hamas through its control of a border crossing from Egypt to the Gaza Strip, Kerry said there was still "work to do" to resolve the conflict.

Israel pounded targets in the Gaza Strip on Tuesday and Justice Minister Tzipi Livni, seen as the most dovish member of Prime Minister Benjamin Netanyahu's inner security Cabinet, told Israeli Army Radio: "A cease-ire is not near."

More than 600 people have died since Israel launched an offensive on July 8 to halt missile attacks out of the Gaza Strip by Hamas, which has been angered by a crackdown on its supporters in the occupied West Bank.

"Hamas has a fundamental choice to make, and it is a choice that will have a profound impact for the people of Gaza," Kerry told reporters in a joint appearance with Egyptian Foreign Minister Sameh Shukri.

"The Egyptians have provided a framework and a forum for them to be able to come to the table to have a serious discussion together with other factions of the Palestinians," he added, calling the U.S.-Egyptian talks "constructive meetings".

Hamas has already rejected the Egyptian initiative, saying the plan ignores its demands for the release of prisoners and for a more comprehensive lifting of an economic blockade on the Gaza Strip imposed to stop arms smuggling.

The Egyptian plan does not specify a timeline for easing the blockade, saying "crossings shall be opened and the passage of persons and goods through border crossings shall be facilitated once the security situation becomes stable on the ground."

In a sign of the intensity of the U.S. diplomacy, Kerry spoke to Netanyahu and to the foreign ministers of Turkey and Qatar after his two-hour meeting with Sisi, according to a senior U.S. official.

Because the United States views Hamas as a terrorist group, U.S. diplomats have no direct contact and Washington must rely on proxies such as Egypt, Qatar and Turkey.

U.S. officials view Qatar, a tiny, gas-rich Gulf state that has supported Hamas financially and hosts some of the militant group's senior leaders, as important to the diplomacy because of its ties to the Palestinian group.

In contrast, the current Egyptian government is deeply suspicious of Hamas because it is an offshoot of the Muslim Brotherhood, the Islamist movement toppled from power in Cairo by Sisi last year when he was chief of the army..

Palestinian President Mahmoud Abbas's Western-backed Fatah movement has also proposed a formula for ending the fighting, calling for an immediate cease-fire followed by five days of negotiations, Palestinian official Azzam al-Ahmed said in Cairo.

The upcoming Eid al-Fitr festival, Islam's biggest annual celebration that follows the end of the fasting month of Ramadan this weekend, could provide all sides with a convenient moment to agree to a cease-fire..