* Palestinians say 341 killed in fighting; 5 Israelis dead
* Israel says half of Hamas's rockets used or destroyed
* U.N.'s Ban to meet Abbas in Qatar, tour region in truce bid
By Nidal al-Mughrabi and Maayan Lubell
GAZA/JERUSALEM, July 20 (Reuters) - Israel fired shells into the Gaza Strip and militants kept up rocket fire into the Jewish state on Sunday with no sign of a diplomatic breakthrough to end the worst fighting between Israel and Hamas in two years.
Israeli tank shells killed one Palestinian and hit houses in the northern Gaza district of Shejaia, where residents called local radio stations pleading for evacuation. Shelling killed four Palestinians near the southern town of Rafah, health officials said.
Explosions rocked through Gaza overnight and shells fired by Israeli naval forces lit up the sky. In Israel, sirens sounded in towns near Gaza, warning of approaching rockets.
Israel sent ground forces into Hamas-dominated Gaza on Thursday after 10 days of heavy air and naval barrages failed to stop rocket fire from the Palestinian territory.
Gaza officials said at least 341 Palestinians, many of them civilians, have been killed in the 13-day conflict. On Israel's side, three soldiers and two civilians have died.
The land incursion has so far failed to subdue Hamas and its allies. They fired more than 90 rockets into Israel on Saturday, the Israeli army said. Israel has vowed to destroy a network of tunnels out of Gaza and hunt down the militants' stockpiles of missiles.
Diplomatic efforts to secure a ceasefire involving, among others, Egypt, Qatar, France and the United Nations, have failed to make headway.
Qatar was due to host a meeting between Palestinian President Mahmoud Abbas and U.N. Secretary-General Ban Ki-moon on Sunday, a senior Qatari source told Reuters. Ban was due during the week to travel to Kuwait, Egypt, Israel, the Palestinian Territories and Jordan, a U.N. statement said.
The Qatari source said Abbas was also due to meet with Hamas leader Khaled Meshaal.
Western-backed Abbas in April struck a deal with Islamist Hamas that led to the formation of a Palestinian unity government, seven years after the group seized control of Gaza from Abbas's Fatah party. Israel responded by suspending U.S.-sponsored peace talks.
Hamas has rejected Egyptian efforts to end fighting, saying any deal must include an end to a blockade of the coastal area and a recommitment to a ceasefire reached after an eight-day war in Gaza in 2012.
Egypt said on Saturday it had no plans to revise its ceasefire proposal. A Hamas source in Doha said the group has no plans to change its conditions to ceasefire.
Israel is wary of mediation by Qatar, which hosts a large number of exiled Islamists from across the Middle East including Meshaal, and Israeli officials have said Egypt must be a party to any ceasefire deal.
French Foreign Minister Laurent Fabius, who flew to Israel after meetings in Egypt and Jordan, said on Saturday efforts to secure a ceasefire so far had failed.
"On the contrary, there's a risk of more civilian casualties that worries us," Fabius said, after talks with Israeli Prime Minister Benjamin Netanyahu in Tel Aviv.
Hostilities between the two sides escalated following the killing last month of three Jewish students that Israel blames on Hamas. Hamas neither confirmed nor denied involvement.
The apparent revenge murder of a Palestinian youth in Jerusalem, for which Israel has charged three Israelis, further fuelled tensions.
Israeli bulldozers and engineers worked along a 1.5-km-wide (one-mile-wide) strip of Gaza's eastern frontier, uncovering 13 tunnels, at least one of them 30 metres (90 feet) deep, military spokesman Lieutenant-Colonel Peter Lerner said.
About 95 rocket launchers were also found and destroyed in the sweep, he added.
Searches were continuing in what he described as an open-ended mission that had "severely impeded Hamas capabilities".
Israel says more than 1,700 rockets have been fired out of Gaza during this month's fighting, and between 3,000 and 4,000 destroyed in military strikes - together almost half of the militants' original estimated arsenal.
Hamas says it is continuously replenishing its stock of weapons and is ready for a prolonged conflict.
The Israeli death toll has been kept low due to the rockets' relative inaccuracy, a network of air raid sirens and shelters and the Iron Dome rocket interceptor's 90 percent success rate.
The Israeli military urged Palestinians to flee a growing area of Gaza ahead of further military action in the Mediterranean coastal enclave. Locals say about half of the territory's 1.8 million population have been told to move.
With both the Israeli and Egyptian borders sealed off, Gazans say they have few places to escape to.
The largest United Nations agency in Gaza, UNRWA, said about 61,500 people had sought refuge in its buildings, mainly schools - more than in any previous conflict there between Israel and Islamist militants. (Writing by Maayan Lubell; Editing by Mohammad Zargham)