(Reuters) - Iraqi troops aided by Shi'ite paramilitaries have driven Islamic State out of central Tikrit, Prime Minister Haidar al-Abadi said on Tuesday, but the fight to retake all of Saddam Hussein's hometown continued.

Government forces have been in a month-long fight for the city, which became a bastion for the Sunni jihadists who are at war with Baghdad and have been targeted by U.S.-led air strikes.

Hundreds of insurgents ready to fight to the death are still holed up in Salahuddin province's capital city and at least three neighborhoods remain under Islamic State control, along with a palace complex in the city's north.

The further Iraqi forces push into the city, the greater the risk of ambushes.

"Our security forces have reached the center of Tikrit and they have liberated the southern and western sides and they are moving toward the control of the whole city," Abadi said in a statement.

The U.S. Defense Department said it could confirm Iraqi security forces' "advancement into Tikrit to liberate the city center as well as other parts of the city" from Islamic State.

In their push from southern Tikrit, security forces and paramilitary fighters retook the governor's headquarters and the main hospital, which had been occupied by Islamic State.

Reuters journalists traveling with the police passed houses scarred by bullets, mortars and rockets as well as five or six corpses that security officers said were Islamic State fighters, adding that they might be rigged with explosives.

Iraqi officials suspect the militants have planted scores of bombs and are using snipers and a network of underground tunnels and bunkers to slow the government advance.

Tuesday saw Shi'ite militia groups return to the battle after suspending operations last Thursday when U.S.-led air strikes were requested by Prime Minister Haidar al-Abadi.

The Iranian-backed armed factions had opposed U.S.-led strikes, insisting that their paramilitary forces could retake Tikrit, seized last June by Islamic State militants as they raced across northern Iraq.

Anti-American groups Kata'ib Hezbollah and Asaib Ahl al-Haq said they had joined federal police and army forces in going deeper into the city on Tuesday after Abadi, a moderate Shi'ite Islamist who became prime minister last year, agreed to halt U.S. air strikes.

They said air strikes in the city on Tuesday were being carried out only by the Iraqi military. A federal police officer made a similar claim.

But the prime minister's office said no such order had been issued and U.S. officials said they were not aware of any military freeze.

The U.S. government, which deeply mistrusts the pro-Iranian Shi'ite militias, has sought ways to participate in the Tikrit battle without acknowledging working with forces backed by Tehran.

U.S. officials have insisted on an Iraqi government military command for the fight, even as Shi'ite militia forces remain the strongest presence on the ground.