Philippine troops engaged Islamic State-linked militants in pitched clashes on Thursday around the southern city of Marawi as the military deployed tanks to secure vital installations in a battle that has become a major test for President Rodrigo Duterte’s government.

With thousands fleeing the city and militants holding a Roman Catholic priest and others hostage, television footage showed black smoke billowing near Marawi’s City Hall as civilians who had held out in the besieged city watched from rooftops.

Explosions and bursts of gunfire could be heard at a distance, as the military asserted it was making progress clearing out “desperate” militants, forcing them to split into groups as they darted around the city trying to flee.

The fighting led Mr. Duterte to impose martial law over the southern Philippines this week. The military’s claims of success in battling the insurgents appeared to clash with images and reports from the city and surrounding areas that suggested a chaotic situation.

“What they are doing now are desperate moves,” Lt. Col. Joar Herrera, a military spokesman, said of the insurgents. “That is why they are engaging in terroristic activities to distract the focus of our military,” he added from an army camp outside the city.

He said troops had started to move around the besieged city of 200,000, extracting residents who were trapped inside their homes when members of the extremist Abu Sayyaf group, backed by the radical Maute gang, attacked on Tuesday.

The military began fighting after receiving reports that Isnilon Hapilon, the leader of an Abu Sayyaf faction who has declared his allegiance to the Islamic State, was spotted in the area. But the government troops were surprised when they encountered a 100-strong Maute force armed with high-powered weapons.

Mr. Herrera said Mr. Hapilon was believed to still be trapped in the area, describing the enemy’s movement as “very fluid.”

“The movement of the local terrorist group is not fixed,” he said. “Our armed forces are constantly repositioning. We need to move so we could totally outstep and outmaneuver them.”

“We have identified targets that we need to clear, and we need to neutralize the remnants of this local terrorist group,” Mr. Herrera said.

Fighting was concentrated in at least three villages on the outskirts of the city, and troops have been moving into the area to clear militants. Helicopters armed with machine guns tried to flush out the insurgents who had taken a bridge in the city, Reuters reported. Mr. Herrera said about 40 rebels were still believed scattered around the city.

There was no word yet on the fate of the priest and his fellow hostages. While the Philippines is Asia’s center of Catholicism, the city of Marawi is overwhelmingly Muslim.

Firefights were described as sporadic on Thursday, and the death toll stood at 13 militants, five soldiers and two police officers. Thirty soldiers were wounded, the military said.

There were also reports of gunmen killing civilians, but these have yet to be verified, according to the army. One police officer was seen lying dead in a police station that was attacked by gunmen.

The intensity of the fighting caught the government by surprise, leading Mr. Duterte to cut short a trip to Russia in addition to declaring martial law in the south.

On Wednesday, he raised the prospect that Islamic State-linked rebels could gain a foothold in the northern Philippines and warned that he might impose military rule over the rest of the country. That alarmed rights groups, which said it would be a throwback to the dictatorial rule of Ferdinand Marcos, whose name Mr. Duterte invoked.

Rights groups have warned that the martial law declaration could lead to more abuses, including more killings in the president’s deadly war on drugs, which has so far left thousands dead since last year.

Mr. Duterte’s “casual reference to the late dictator should be especially alarming,” said James Ross of Human Rights Watch, which is based in New York.

“Duterte faces one significant obstacle to becoming the next Marcos: the 1987 Philippine Constitution, which places restrictions on the imposition and conduct of martial law,” he said, adding that Congress could revoke the martial law proclamation by majority vote and that the Supreme Court could rule on the factual basis for its declaration.

Already, several lawmakers have voiced concern, with the opposition leader Francis Pangilinan vowing to “vigorously oppose a nationwide martial law declaration.”

“While there may be isolated terrorist acts in these areas, there exists no Visayas-wide nor a Luzon-wide invasion or rebellion to merit a declaration of martial law,” Mr. Pangilinan said, referring to the president’s threat to expand military rule to the central and northern islands to cover the entire country.

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