* Gaza death toll almost 550, Israeli death toll rises to 29
* Violence flares in the West Bank
* U.N. chief Ban Ki-moon to meet Israeli, Palestinian leaders
* Kerry in Cairo pushing for ceasefire (Adds Israel says 183 Palestinian militants killed, details)
By Nidal al-Mughrabi and Maayan Lubell
GAZA/JERUSALEM, July 22 (Reuters) - Israeli forces killed 10 Palestinians on Tuesday, maintaining its huge offensive against the Gaza Strip as diplomatic pressure mounted to end the worst violence in five years between Islamist Hamas and Israel.
Plumes of dark black smoke rose into the sky above Gaza, and shells continued to rain down on the coastal enclave, while Hamas and its allies fired 11 rockets across the border into southern Israel, the army said.
With the conflict entering its third week, the Palestinian death toll rose to almost 550, including nearly 100 children and many other civilians, Gaza health officials said.
The Israeli military said it had killed 183 militants.
Israel's casualties also mounted, with the military announcing the deaths of two more soldiers, bringing 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 Israelis civilians have also been killed by Palestinian rocket fire into Israel.
Violence spread to the nearby occupied West Bank, where medics said soldiers shot dead a Palestinian man while dispersing stone-throwing protesters. A Palestinian shot and seriously wounded an Israeli in the Nablus area on Tuesday.
U.S. Secretary of State John Kerry arrived in neighbouring Egypt on Monday, sent by President Barack Obama to seek a ceasefire between Israel and Hamas, which controls Gaza.
U.N. Secretary General Ban Ki-moon, who met in Cairo with Kerry, was scheduled to fly to Israel on Tuesday for talks with Prime Minister Benjamin Netanyahu and to meet Palestinian officials in the West Bank.
Israel Radio reported that Hamas's allies, the Islamic Jihad, had said the sides may agree to a five-hour humanitarian truce on Tuesday to let civilians stock up on vital supplies. Israeli officials said they had no knowledge of this.
Israel has signalled it is not in a 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.
Egypt presented a ceasefire plan last week. Israel accepted, but Hamas rejected it, saying the group had not been consulted. An Israeli official in Washington, who declined to be named, said he wanted Kerry to get Egypt to apply pressure on Hamas.
Hamas has demanded Israel and Egypt lift a blockade on the coastal territory packed with 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.
"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.
Kerry said the United States would provide $47 million in humanitarian aid for Gaza. Washington is "deeply concerned about the consequences of Israel's appropriate and legitimate effort to defend itself," Kerry said. "No country can stand by when rockets are attacking it."
"But always, in any kind of conflict, there is a concern about civilians - about children, women, communities that are caught in it," Kerry said.
Kerry plans to stay in Cairo until Wednesday morning but has no set departure date from the region. Officials said he might travel to Qatar, a Gulf state which has relatively close ties to Hamas and hosts its leader, Khaled Meshaal.
The Health Ministry in Gaza said four people were killed and 70 were wounded on Monday at Al-Aqsa hospital, when an Israeli tank shell slammed into its third floor, which housed operating theatres and an intensive care unit.
The International Committee of the Red Cross issued a statement condemning the shelling "in the strongest terms". It said the hospital came under direct fire at least four times and that life-saving equipment had been severely damaged.
The Israeli military, which has accused Hamas militants of firing rockets from the grounds of Gaza hospitals and seeking refuge there, had no comment.
Fighters from Hamas and its allies have repeatedly tried to infiltrate Israel over the past week through the tunnels, looking to attack villages and army encampments that dot the border area, Israel said.
Netanyahu sent in Israeli ground forces on Thursday to destroy the tunnels and the militants' missile stockpile.
(Additional reporting by Arshad Mohammed and Yasmine Saleh in Cairo and Ali Sawafta in Ramallah; Writing by Allyn Fisher-Ilan and Maayan Lubell; Editing by Ron Popeski, Crispian Balmer and Anna Willard)