AL-ARISH, Egypt, Aug 12 (Tamim Elyan and Yusri Mohamed/Reuters) - Egyptian soldiers killed five Islamist militants after storming their hideout near the border with Israel on Sunday, the latest action in an army campaign to reimpose authority on the region, security sources and eyewitnesses said.

The troops tracked down the militants in the settlement of al-Goura, about 15 km (10 miles) from the frontier, as they searched for jihadists who killed 16 Egyptian border guards and tried to infiltrate Israel a week ago.

The latest clash is part of a security sweep that began on Wednesday and is the biggest military operation in the region since Egypt's 1973 war with Israel. No one has claimed responsibility for killing the border guards.

It is an early test for President Mohamed Mursi - a moderate Islamist elected in June following the overthrow last year of Hosni Mubarak - to prove he can rein in militants whose actions on the border worry Israel. In a speech on Sunday, Mursi pledged to restore order.

I won't sleep my nights restfully until the people of Sinai are calm and settled in their homes, he said. We will continue with all strength. We have what it takes to end these criminal remnants.

Security sources said five people had been killed - three from bullet wounds and two more whose scorched bodies were found in the hut which was burned.

People in the area supplied information that there was a group of unidentified people staying in a makeshift hut. The area was immediately raided. The group opened fire and the police returned fire, one police source said.

A senior police officer earlier said six people had been killed.

In addition to the five dead, one militant was seriously wounded and taken to hospital in al-Arish in north Sinai.

The police sources, who did not give their names because they are not authorised to speak to the media, said troops had found guns, rocket launchers, a truck and a motorcycle at the scene.

One officer said among the five dead was a Palestinian, but the report could not be independently confirmed.

A resident of al-Goura told Reuters he had seen the lifeless bodies of two men who were not from the area.

They resisted very strongly, he said by telephone after the clash. They fired rocket-propelled grenades at the troops.

ISRAELI CONCERNS

Searching another area, al-Kharouba, security forces found mortars and other weapons, said a third police source. Police said the operation to sweep the area was continuing and five men had been arrested.

The unidentified assailants who killed the border guards last week stormed through a border crossing before they were killed by Israeli forces.

This prompted Israeli calls for Egypt to reassert control over an increasingly lawless Sinai and a wave of anger among Egyptians, some of it directed at Egypt's new president, who promised to restore security in the region.

Critics say Mursi risks being soft on jihadist groups because he is from the Muslim Brotherhood, a political Islamist movement that has ties to the Hamas government in Gaza and a history of hostile rhetoric towards Israel.

The Brotherhood renounced violence as a means to achieve political change in Egypt decades ago.

The campaign that I am leading myself with the police and armed forces is not against the peaceful or the noble people of Sinai, Mursi said to mark the Muslim fasting month of Ramadan.

The military crackdown began in earnest on Wednesday after unidentified men attacked several checkpoints in al-Arish.

Details of the campaign have been patchy so far because the operation is spread over a wide, under-populated area and some of its inhabitants are reluctant to give details of the operation for fear of official reprisals.

The government in Cairo has brought armored vehicles, tanks and hundreds of extra troops into Sinai but its task is complicated by suspicion of the authorities among local Bedouin who often carry weapons.

Further south from al-Goura, in central Sinai, unidentified gunmen shot at a checkpoint overnight and clashed with security forces there, Egyptian and international officials said. (Reporting by Yusri Mohamed, Tamim Elyan, Yasmine Saleh and Tom Pfeiffer; Writing by Tom Pfeiffer and Edmund Blair; Editing by Angus MacSwan and Alessandra Rizzo)