* First reported use of drone by Palestinian militants
* Mediation offers mount; Israel, Hamas say fight not over
* No sign ceasefire near after week of violence (Updates Palestinian death toll, 2 more Israelis injured, Syria front incident, quotes)
By Nidal al-Mughrabi and Jeffrey Heller
GAZA/JERUSALEM, July 14 (Reuters) - Palestinian militants resumed rocket attacks on Tel Aviv on Monday after a 24-hour lull in strikes on the Israeli commercial capital, and Israel kept up its air and naval bombardments of the Gaza Strip despite growing international pressure for a ceasefire.
The military said it had shot down a drone from Gaza, the first reported deployment of an unmanned aircraft by Palestinian militants and a possible step up in the sophistication of their arsenal, although it was not clear whether it was armed.
Their rocket attacks have been regularly intercepted but more than half a dozen Israelis have been wounded since the start of the week-old offensive. Gaza health officials say Israeli air strikes have killed 170 Palestinians, most of them civilians.
With international calls mounting for a ceasefire, Egyptian media said U.S. Secretary John Kerry was due in Cairo on Tuesday for talks on the Gaza situation. There was no immediate U.S. confirmation of the report.
The European Union said it was in touch with "all parties in the region" to press for an immediate halt to the hostilities, the worst flare-up of Israeli-Palestinian violence for almost two years.
Two members of Prime Minister Benjamin Netanyahu's security cabinet suggested a truce might be in the works. "There are contacts all the time," Communications Minister Gilad Erdan said when asked about the possibility of Kerry visiting Cairo.
Economics Minister Naftali Bennett said in remarks on Channel 2 television "all options are on the table," though the far-right leader said the fighting could also go on.
The violence was prompted by the murder of three Israeli teenagers and revenge killing of a Palestinian youth. Israeli officials said on Monday three people arrested over the Palestinian's death had confessed to burning him alive.
Egypt and Qatar are seen as potential mediators but peace efforts were complicated by Hamas's rejection of a mere "calm for calm" in which both sides hold their fire in favour of wider conditions including prisoner release and an end to Israel's Gaza blockade.
The Israeli army said its aircraft and naval gunboats attacked dozens of targets in the Gaza Strip and that Palestinian militants fired more than 20 rockets into Israel, slightly wounding a boy in the town of Ashdod, where a home was damaged, and two girls, 11 and 13, near Beersheba.
Palestinian health officials said at least 20 people in Gaza were wounded.
But Israel did not carry out a threat to step up attacks against rocket-launching sites it said were hidden among civilian homes in the town of Beit Lahuiya after urging residents there to leave. A U.N. aid agency said around a quarter of the town's 70,000 residents had fled.
Tel Aviv experienced a rare lull in morning rocket strikes, but they resumed during the evening rush hour, with the Iron Dome missile interceptor system going into action. Police said there were no casualties or damage. Hamas, the Islamist group which runs Gaza, said its armed wing had sent several locally-made drones to carry out "special missions" deep inside Israel.
A military spokesman said the drone was shot down near the port of Ashdod, about 25 km (15 miles) north of Gaza, by a U.S.-built Patriot missile, used largely ineffectively by Israel against Iraqi Scud missiles in the 1991 Gulf War.
An Egyptian-brokered truce doused the last big Gaza flare-up in 2012, and U.N. Secretary General Ban Ki-moon told Egyptian President Abel Fattah al-Sisi in a phone call that his country is the most credible party capable of persuading both sides to stand down, an official Egyptian statement said.
But Cairo's government is at odds with Islamist Hamas, complicating a mediation bid with the group, an offshoot of the now-outlawed Egyptian Muslim Brotherhood.
Asked if Egypt was mediating, Foreign Ministry spokesman Badr Abdelatty said only that Cairo was "in close contact with the Israelis and all Palestinian factions as well as with regional and international countries".
He said he did not want to predict whether those efforts were moving Israel and Hamas close to a ceasefire.
A Hamas politbureau member said Kerry called the foreign minister of Qatar this week, asking him to mediate with the Palestinian movement. A Qatari government source said, however, that Hamas had unrealistic conditions for a ceasefire.
"Qatar is the only one that reached out to us," Hamas official Ezzat al-Rishq said in Doha. "I wouldn't say it's mediation - it's still too early - they have just opened a line of communication with us, but there is no clear plan on what form of mediation this will be."
In Cairo, Palestinian sources said they had plans to submit a draft resolution calling for an immediate ceasefire via the Arab League, to the United Nations Security Council.
Israel's Erdan told Channel 2 television Israel wanted "long term calm, and a reality change, for Hamas and terrorist infrastructure to sustain as big a blow as possible". But "whether this happens alongside a diplomatic solution, at some point, we certainly wouldn't rule it out," Erdan said.
SEVENTH DAY OF WARFARE
Al-Mezan, a Gaza-based Palestinian human rights group, said 869 Palestinian homes have been destroyed or damaged in Israeli attacks over the past week.
Hamas leaders have said a ceasefire must include an end to Israel's Gaza blockade and a recommitment to the 2012 truce agreement. In addition, Hamas wants Egypt to ease restrictions it imposed at its Rafah crossing with the Gaza Strip since the military toppled Islamist president Mohamed Mursi last July.
Hamas has faced a cash crisis and Gaza's economic hardship has deepened as a result of Egypt's destruction of cross-border smuggling tunnels. Cairo accuses Hamas of aiding anti-government Islamist militants in Egypt's Sinai peninsula, an allegation the Palestinian group denies.
For its part, Hamas leaders said, Israel would have to release hundreds of the group's activists it arrested in the occupied West Bank last month while searching for the three Jewish seminary students who it said were kidnapped by Hamas.
The detainees include more than 50 Hamas men who were freed from Israeli jails in 2011 in a prisoner exchange.
Israel's Gaza offensive, which began last Tuesday, has claimed the lives of at least 138 Palestinian civilians, including 30 children, health officials in the enclave said.
In all 170 Palestinians have been killed, the latest in an air strike targetting a motorcycle in Gaza, the Palestinian Health Ministry said.
There have been no fatalities in Israel in the fighting. Iron Dome, which is Israeli-built and partly funded by the United States, has intercepted many of the rocket salvoes.
But the persistent rocket fire has disrupted life in major cities, paralyzed vulnerable southern towns and triggered Israeli mobilisation of troops for a possible Gaza invasion if the Palestinian rockets persisted.
In violence along the Syrian frontier, a rocket fired at the Golan Heights, territory Israel captured from Syria in a 1967 war and later annexed, caused a fire but no casualties.
(Additional reporting by Maayan Lubell and Dan Williams in Jerusalem, Michael Georgy in Cairo and Amena Bakr in Doha; writing by Jeffrey Heller and Allyn Fisher-Ilan; editing by Philippa Fletcher)