AL-ARISH, Egypt, Aug 9 (Reuters) - Gunman fired shots towards a police station in the main administrative centre of Egypt's North Sinai on Thursday, underscoring lawlessness in the desert region bordering Israel as a Egyptian military offensive there entered its second day.

Hundreds of troops in armoured cars drove out of the town to hunt Islamist militants blamed for killing 16 Egyptian border guards on Sunday, the biggest spike in violence which has been growing steadily since last year's overthrow of Hosni Mubarak.

The gunfire in al-Arish, the nerve centre of the government's otherwise shaky control of the North Sinai region, showed how difficult it will be for Egypt to impose order. It followed attacks on checkpoints in the town on Wednesday.

Israel has welcomed Egypt's offensive while continuing to express worries about the deteriorating situation in Sinai, home to anti-Israel militants, Bedouin tribes angered by neglect by Cairo, gun-runners, drug smugglers and al Qaeda sympathisers.

Israeli Defence Minister Ehud Barak said Egypt was acting "to an extent and with a determination that I cannot previously recall".

"Whether this ends with (their) regained control of Sinai and allows us not to worry as much as we have in the past few months, this I do not know," he told Israel Radio.

The unidentified gunmen in al-Arish fled before police could respond, a security source said, denying a report by state television that police had fought back.

Hundreds of troops and dozens of military vehicles had reached the town, security sources said, part of an offensive not seen since Egypt's 1973 war with Israel.

Dozens of armoured vehicles, some equipped with machine-guns, could then be seen driving out of al-Arish towards the settlement of Sheikh Zuwaid which military aircraft attacked on Wednesday. The troops saluted passers-by and flashed victory signs, or filmed their departure with video cameras.

PUBLIC ANGER

Egypt's President Mohamed Mursi - whose Islamist background in the Muslim Brotherhood has been eyed with suspicion by Israel since he was elected in June - on Wednesday fired the region's governor and country's intelligence chief in response to public anger over Sunday's attack.

No one has claimed responsibility for the assault, in which the assailants seized two armoured vehicles to storm an Israeli border crossing. One made it through before the attackers were killed by Israeli fire.

Israel says militants based in Sinai and Palestinian hardliners in neighbouring Gaza pose a growing threat to its border. It says Palestinians use illegal tunnels to smuggle in guns and travel across to join those on the Egyptian side.

Israel has also been wary of Mursi's ideological affinity with Hamas, the Islamist group ruling Gaza, fearing he would take a softer position on Palestinian militancy than Mubarak.

Mursi has brushed aside accusations that his politics would make it difficult for him to take a strong stance against violent groups sworn to Israel's destruction.

His response to Sunday's attack, which happened during the evening "iftar" meal which breaks the daytime fast during the Muslim holy month of Ramadan, may also be underpinned by public anger over the deaths of the border guards.

In Egypt, there is wide respect for rank-and-file soldiers who are often poorly paid conscripts working in isolated places far from their families.

Comments suggesting outgoing intelligence chief Mourad Mwafi had been aware of a threat but took no action fuelled that anger - despite suggestions he had been used as a scapegoat.

"...we never imagined that a Muslim would kill his Muslim brother at iftar," Egypt's state news agency MENA quoted Mwafi as telling his Turkish counterpart.

Mursi's powers, are in any case, hemmed in by the army, which retains a strong role in setting security policy.

Residents in al-Arish, meanwhile, welcomed the security sweep, seeing it as an opportunity to curb criminality among Bedouin tribes, including those in Sheikh Zuwaid, who make their living smuggling goods and people through a network of more than 1,000 tunnels into Gaza.

"We want the army to return to the border," said 45-year-old shopkeeper Hassan Mohamed. "The tunnels have destroyed the lives of people in al-Arish. We want them to hit the Bedouin hard."