BAGHDAD — An assault by Iraqi forces to wrest control of Ramadi from the Islamic State reached the edges of the city center Tuesday evening, in a battle that was months in the making and a critical test for the Iraqi government.

Accompanied by heavy US airstrikes, the push into the city by a mix of Iraqi soldiers, police officers, and Sunni tribal fighters began overnight. Iraqi officials described the assault as a fierce urban battle, with their forces facing car bombs, sniper fire, and explosive traps. Around 300 Islamic State fighters are believed to be hunkered down in the northern reaches of the city.

If Iraqi forces manage to reassert control over Ramadi — the provincial capital of Anbar province, in the Sunni Arab heartland — it would be the most important of a series of military setbacks for the Islamic State since its explosive expansion across Iraq that began with the capture of Mosul last year.

In early April this year, Iraqi forces and Shi’ite militias drove the Islamic State out of the city of Tikrit, and in October they retook control of the northern city of Baiji and its oil refinery. Last month, Kurdish and Yazidi forces assaulted the northern city of Sinjar, driving out fighters with the Islamic State, also known as ISIS or ISIL.

The capture of Ramadi, 60 miles from Baghdad, would give the government of Prime Minister Haider al-Abadi a badly needed moral victory, and a successful cooperation with the country’s alienated Sunni population. But more important, it would allow his often disparaged military to reverse a humiliating loss.

Ramadi fell to the Islamic State in May, in a sudden collapse after a long battle that exposed multiple weaknesses in the government’s ability to fight the militants, including stark military shortfalls and disorganization, and an unwillingness by the government to arm or send reinforcements to help Sunni tribesmen who were fighting the militants.

The rapid advances Monday and Tuesday held out hope that after months of preparation, the government had finally marshaled a large enough force to prevail in Ramadi, and begin a wider operation to fight the Islamic State in other areas of Anbar province.

“I think the fall of Ramadi is inevitable,’’ Colonel Steven H. Warren, the US military spokesman here, said Tuesday. But he added: “That said, it’s going to be a tough fight.’’

Other crucial battles, like the ones for Tikrit and Baiji, dragged on for weeks or months, and it remained to be seen whether the Islamic State would quickly melt away.

Over the past month or so, Iraqi security forces and tribal fighters have encircled the city. Two weeks ago, they seized a large neighborhood, Tamim, on its southwestern outskirts.

“We went into the center of Ramadi from different axes, and we started clearing residential areas,’’ General Sabah al-Numani, a spokesman for the army counterterrorism unit in charge of the offensive, said in a statement Tuesday. He predicted that “the city will be cleared within the coming 72 hours.’’

Six hundred to 1,000 Islamic State fighters were said to have been in Ramadi when the offensive began two weeks ago, but several hundred of them have been killed in heavy fighting since then, according to Iraqi and Western officials.

Those remaining did not appear to be giving up easily. They destroyed three bridges over the Euphrates River to prevent security forces from entering the city, according to General Ahmed al-Belawi, the leader of a battalion of Sunni tribal fighters.

Warren said that Iraqi forces had crossed the river by deploying “floating bridges’’ capable of moving fighters and heavy equipment across the water, as US troops had trained them to do.

Al-Jazeera reported that 14 soldiers and 17 tribal fighters were killed by a suicide car bomber in Albu Diab, northwest of the city center, and that at least 12 militants had been killed. MSNBC released a video that it said showed an Islamic State counterattack on the eastern edge of the city, and it quoted a tribal fighter as saying that at least seven Islamic State militants had been killed. Those casualty numbers could not be independently confirmed.

Iraqi airplanes dropped leaflets Sunday urging residents of Ramadi to evacuate within 72 hours, warning of an impending operation, and suggesting two evacuation routes. Warren estimated that thousands or even tens of thousands of civilians were still in the city; hundreds of thousands of others have fled.

In a telephone briefing Tuesday, Warren said that coalition forces had recovered what he said were Islamic State leaflets in the nearby city of Fallujah urging its fighters — if they lose control of the city — to impersonate Iraqi security forces and commit atrocities.

The authenticity of the leaflets could not be independently confirmed.