BAGHDAD — Iraqi forces pushed into the southern part of Fallujah on Monday, encountering rockets and sniper fire from Islamic State fighters determined to hold onto the strategic western city.

Lieutenant General Abdulwahab al-Saedi, commander of the government’s offensive, said that forces from Iraq’s army, police, and elite Counterterrorism Service launched the assault on southern Fallujah early Monday. They were backed by planes from the US-led coalition.

Since Prime Minister Haider al-Abadi announced the operation just over a week ago, government forces, together with Shi’ite militia troops and tribal fighters, mostly from Sunni Anbar province, have taken territory around Fallujah in preparation for an assault on the city.

Capturing Fallujah, the first Iraqi city to fall to the Islamic State in 2014, is expected to be both an important battle and a difficult one.

The city, about 45 miles west of Baghdad, has a history of nurturing armed Islamist groups. In 2004, US Marines fought two fierce battles to take control of Fallujah, but insurgents returned each time.

The advance of the Iraqi troops is likely to be slow because tens of thousands of civilians remain trapped in Fallujah and hidden bombs are believed to have been left throughout the city, according to special forces commanders.

They expect fierce resistance from the jihadis. The 500 to 700 Islamic State fighters holed up in Fallujah are believed to be among the group’s best-trained

Fallujah, a predominantly Sunni city in Anbar province, is one of the last major Islamic State strongholds in Iraq. The extremist group still controls territory in the north and west, as well as the second-largest city of Mosul.

Saedi said that as government forces approached the Shuhadaa neighborhood in southern Fallujah, the militants detonated at least two car bombs and were using snipers and rockets to try to fend off the advancing troops.

‘‘The Counterterrorism Service forces have the experience to fight them, just like they did in Ramadi, Hit, and Rutba,’’ Saedi said, referring to other recent battles.

Shi’ite militia forces under the government umbrella of the Popular Mobilization Forces and the federal police have led operations that have taken back 80 percent of the territory around Fallujah, the Associated Press reported, quoting Iraqi Major Dhia Thamir.

North of Fallujah, federal police, soldiers, and militiamen pressed into the town of Saqlawiya, while police SWAT forces raised the Iraqi flag over the police station in Na’imiyah, south of the city.

The Iraqi military also said its warplanes had launched attacks on Islamic State positions around Fallujah on Monday. Colonel Steve Warren, a US military spokesman, said that American aircraft were also conducting strikes.

In and around Baghdad on Monday, a wave of bombings claimed by the Islamic State killed at least 24 people, the AP reported.

The deadliest of the blasts took place in the northern, Shi’ite-dominated Shaab neighborhood of Baghdad, where a suicide bomber rammed his explosives-laden car into a checkpoint next to a commercial area, killing eight civilians and three soldiers. The explosion also wounded up to 14 people, a police officer said.

A suicide car bomber struck an outdoor market in the town of Tarmiyah, about 30 miles north of Baghdad, killing seven civilians and three police officers.

In an online statement, the Islamic State said it was responsible for the attacks, saying it targeted members of the Shi’ite militias and a government office.

Officials believe a recent surge in terrorist bombings in the Iraqi capital reflects an effort by the Islamic State to divert government attention from the battlefield. Authorities are hoping that recapturing Fallujah will reduce that threat.

Lieutenant General Abdul Ghani al-Assadi, commander of the Counterterrorism Service, said Fallujah was an especially complicated operation because of its history of supporting insurgents and because the battle would unfold in an urban area with a significant civilian population.

Lieutenant General Sean MacFarland, the US commander in Iraq, warned last week that government forces may have to contend with Islamic State sympathizers among the local population.

Reasserting government control of Fallujah would deal a major blow to the group in western Iraq, where it has been able to draw strength from fellow militants in neighboring Syria.

A government victory also would help build momentum ahead of a highly anticipated operation to recapture Mosul to the north.

The Counterterrorism Service troops are expected to play a key role in leading the charge into the heart of Fallujah, as they have in other battles.

Assadi, speaking in an interview on Sunday, said those elite forces had gradually figured out the Islamic State’s way of waging war, a combination of conventional military tactics and insurgent attacks.

Saedi said security forces had chosen to attack from southern Fallujah because that area was largely empty of civilians. About 50,000 civilians are believed to remain in besieged Fallujah, where conditions have worsened as food and other basic supplies grow scarce.

Mayor Issa al-Issawi, speaking from outside the city, said civilians had begun to move from southern areas of Fallujah toward the north, and may be used by militants as human shields.

He said there had been reports of civilian casualties from recent shelling and air operations by government-aligned forces. Some residents have been killed by militants, meanwhile, while trying to flee, officials from the city have said. The Post could not independently confirm those reports.

The Iraqi military, in a statement Monday, said counterterrorism troops in the Fallujah area had killed an Islamic State fighter named Abu al-Harith, suspected of manufacturing car bombs there.