* Gaza death toll tops 600, Israeli death toll rises to 29
* FAA bans U.S. carriers from flying to Israel
* U.N. chief Ban Ki-moon to meet Israeli, Palestinian leaders
* Kerry in Cairo pushing for ceasefire (Adds details on flight suspensions, tunnels uncovered, death toll)
By Nidal al-Mughrabi and Jeffrey Heller
GAZA/JERUSALEM, July 22 (Reuters) - Israel pounded targets across the Gaza Strip on Tuesday, saying no ceasefire was near as top U.S. and United Nations diplomats pursued talks on halting the fighting that has claimed more than 600 lives.
U.S. Secretary of State John Kerry held discussions in neighbouring Egypt, while U.N. Secretary General Ban Ki-moon met Prime Minister Benjamin Netanyahu in Tel Aviv and later with the Palestinian prime minister in the occupied West Bank.
However, there was no let-up in the fighting around Gaza, with plumes of black smoke spiralling into the sky, and Israeli shells raining down on the coastal Palestinian enclave.
Dealing a blow to Israel's economy already reeling from a spate of tourism cancellations, the Federal Aviation Administration (FAA) took the rare step of banning U.S. carriers from flying to or from Ben-Gurion International Airport for at least 24 hours after a rocket fired from Gaza struck near the airport's fringes, injuring two people.
European airlines including Germany's Lufthansa, Air-France, Dutch airline KLM, Norwegian Air SAS and Turkish Airlines, said they were halting flights there too. Israel's flagship carrier El Al continued flights as usual.
Israel launched its offensive on July 8 to halt missile salvoes out of the Gaza Strip by Hamas, the dominant group in the coastal territory, which was angered by a crackdown on its supporters in the occupied West Bank and suffering economic hardship because of an Israeli-Egyptian blockade.
"A ceasefire is not near," said Justice Minister Tzipi Livni, viewed as the most dovish member of Netanyahu's inner security cabinet. "I see no light at the end of the tunnel," she told Israel's Army Radio.
Dispatched by U.S. President Barack Obama to the region to seek a ceasefire, Kerry met on Tuesday with Egyptian President Abdel Fattah al-Sisi and Foreign Minister Sameh Shukri.
"There is a framework ... to end the violence and that framework is the Egyptian initiative," Kerry said at a news conference with Shukri.
"For the sake of thousands of innocent families whose lives have been shaken and destroyed by this conflict, on all sides, we hope we can get there as soon as possible," he said.
"Hamas has a fundamental choice to make and it is a choice that will have a profound impact for the people of Gaza."
Egypt was key to securing an end to a previous bout of Gaza fighting in 2012, but the country's new leadership is openly hostile to Hamas. For its part, Hamas has already rejected Egypt's proposal this time around as insufficient.
"We hope (Kerry's) visit will result in a ceasefire that provides the necessary security for the Palestinian people and that we can commence to address the medium and long-term issues related to Gaza," Shukri said.
With the conflict entering its third week, the Palestinian death toll rose to 621, including nearly 100 children and many other civilians, Gaza health officials said.
The latest strikes killed a six-month-old infant, a 62-year-old man and three other men riding on motorcycles in southern Gaza, Palestinian health officials said. The Israeli military said it had killed at least 183 militants.
Israel's casualties also mounted as the military announced the deaths of two more soldiers. That brought the number of army fatalities to 27, almost three times as many as were killed in the last ground invasion of Gaza in a 2008-2009 war.
Two Israeli civilians have also been killed by Palestinian rocket fire into Israel.
Addressing reporters with Netanyahu at his side, U.N. chief Ban said: "My message to Israelis and Palestinians is the same: Stop fighting. Start talking. And take on the root causes of the conflict, so we are not back to the same situation in another six months or a year."
Kerry has said the United States would provide $47 million in humanitarian aid for the Gaza Strip. He plans to stay in Cairo until Wednesday morning but has no set departure date from the region.
An Egyptian official who attended some of Kerry's meetings said Ban was working toward reaching a humanitarian truce, perhaps lasting for several days, to get aid into the territory.
"The sensitivities between Egypt and Hamas are what is halting a final inclusive ceasefire deal," the official said.
Palestinian President Mahmoud Abbas's Western-backed Fatah movement also proposed a formula for ending the fighting, calling for an immediate ceasefire followed by five days of negotiations, Palestinian official Azzam al-Ahmed said in Cairo.
Abbas said Palestinian "anger is great" against the Israeli offensive and pledged in his remarks from Ramallah to "go everywhere to stop the aggression and chase down all those who committed crimes against our people."
With Israeli shells and bombs hitting Gaza day and night, thousands of people have fled districts close to the border. The main U.N. agency in Gaza, UNRWA, said almost 102,000 people had taken shelter in 69 of its schools.
UNRWA said it found rockets hidden in a vacant Gaza school near two buildings housing refugees, in the second such instance of militants accused of storing weaponry in a school during the latest offensive.
An UNRWA statement said staff were removed from the building where the rockets were found adding that it "strongly and unequivocally condemns the group or groups responsible".
ISRAEL IN NO HURRY
Israel has signalled it is in no hurry to achieve a truce before reaching its goal of crippling Hamas's militant infrastructure, including rocket arsenals and networks of tunnels threatening Israelis living along the Gaza frontier.
Lieutenant Colonel Peter Lerner, a military spokesman, said Israel had eliminated about half of the tunnels Hamas had dug with the aim of infiltrating into Israel, and destroyed 30 to 40 percent of militants' rocket arsenals in Gaza.
Israel said Gaza militants had fired 2,160 rockets at Israel since the start of the offensive and about a fifth of them had been intercepted by the Iron Dome. Eighty-seven rockets were fired at Israel on Tuesday and 18 of them intercepted.
Hamas has said it will not cease hostilities until its demands are met, including that Israel and Egypt lift their blockade of Gaza and its 1.8 million people, and that Israel release several hundred Palestinians detained during a search last month for three Jewish teenagers later found dead.
Israel blamed the killings on Hamas, and their deaths, along with the revenge slaying of a Palestinian teen, were factors in a flare-up of violence along the Israel-Gaza border last month that escalated into the current fighting.
"The world must understand that Gaza has decided to end the blockade by its blood and its heroism," deputy Hamas leader Ismail Haniyeh said in a televised address on Monday.
Livni said the Hamas demands were unacceptable to both Israel and Egypt.
In Israel, the military said it had identified the remains of six soldiers killed in an attack on their armoured vehicle in Gaza on Sunday and was trying to identify the seventh.
Prompting widespread celebrations in Gaza, Hamas's armed wing announced on Sunday that it had captured a soldier. It displayed a photo ID and army serial number of the man, but did not show any image of him in their hands.
The Israeli military believes it was impossible for anyone to have survived the direct hit on the armoured vehicle in which the missing man was travelling.
Israel has agreed to mass releases of Palestinian prisoners in the past to secure the freedom of captured soldiers, or even for the return of the bodies of its citizens. (Additional reporting by Noah Browning in Gaza, Arshad Mohammed, Shadia Nasralla and Yasmine Saleh in Cairo, Ali Sawafta in Ramallah and Maayan Lubell in Jerusalem; Writing by Jeffrey Heller and Crispian Balmer; Editing by Giles Elgood, Ruth Pitchford and Tom Heneghan)