Suicide bombers who carried out deadly attacks on three churches in Indonesia's second-largest city on Sunday were a family of six that included two young children, police said, as the world's most populous Muslim nation recoiled in horror at one of the worst attacks on its Christian minority.

At least seven people plus the six bombers died in the attacks in Surabaya, according to police.

At least 41 people were injured in the attacks, which Indonesia's president condemned as “barbaric.”

The bombings were the worst to target churches in Indonesia since a series of attacks on Christmas Eve in 2000 killed 15 people and wounded nearly 100.

Religious minorities in Indonesia, especially Christians, have been repeatedly targeted by militants.

National police chief Tito Karnavian said that the father exploded a car bomb, two sons aged 18 and 16 used a motorcycle for their attack, and the mother was with daughters aged 12 and 9 for her attack.

Karnavian said the family had returned to Indonesia from Syria, where until recently the Islamic State group controlled significant territory.

The extremist group claimed responsibility for Sunday's attacks in a statement carried by its Aamaq news agency. It didn't mention anything about families or children taking part in the attack, and said there were only three attackers.

The first attack struck the Santa Maria Roman Catholic Church in Surabaya, police spokesman Frans Barung Mangera told reporters at the scene. That blast was followed by a second explosion minutes later at the Christian Church of Diponegoro and a third at the city's Pantekosta Church, Mangera said.

A witness described the woman's attack at the Diponegoro church, saying she was carrying two bags when she arrived.

“At first officers blocked them in front of the churchyard, but the woman ignored them and forced her way inside. Suddenly she hugged a civilian then (the bomb) exploded,” said the witness, a security guard who identified himself as Antonius.

Mangera said three unexploded homemade bombs, two at the Pantekosta church and one at the Diponegoro church, were detonated by a bomb squad.

Shattered glass and chunks of concrete littered the entrance of the Santa Maria Church, which was sealed off by armed police. Rescue personnel treated victims at a nearby field while officers inspected wrecked motorcycles in the parking lot that had been burned in the explosion.

A street merchant outside the church said she was blown several metres (yards) by the blast.

“I saw two men riding a motorbike force their way into the churchyard. One was wearing black pants and one with a backpack,” said the merchant, Samsia, who uses a single name. “Soon after that the explosion happened.”

President Joko “Jokowi” Widodo visited the scenes of the attacks and described them as “cowardly actions” that were “very barbaric and beyond the limit of humanity.”

In Jakarta, Indonesia's capital, the Indonesian Church Association condemned the attacks.

“We are angry,” said Gormar Gultom, an official with the association, but urged people to let the police investigation take its course.

Indonesia's two largest Muslim organisations, Nahdlatul Ulama and Muhammadiyah, also condemned the attacks.

Separately, national police spokesman Setyo Wasisto said police fatally shot four suspected militants and arrested two others early on Sunday in West Java towns. It wasn't clear whether the shootings were connected to the church attacks.

“They have trained in order to attack police,” Wasisto said, identifying the militants as members of Jemaah Anshorut Daulah, an Islamic State group-affiliated network of about two dozen extremist groups that has been implicated in a number of attacks in Indonesia over the past year.

Jakarta police placed the capital and surrounding areas on high alert, while the transportation ministry warned airports to be on guard.

The church attacks came days after police ended a hostage-taking ordeal by imprisoned militants at a detention center near Jakarta in which six officers and three inmates died. IS claimed responsibility.

Indonesia has carried out a sustained crackdown on militants since bombings by Al Qaeda-affiliated radicals in Bali in 2002 killed 202 people. In recent years, the country has faced a new threat as the rise of IS in the Middle East invigorated local militant networks.

Christians, many of whom are from the ethnic Chinese minority, make up about 9 per cent of Indonesia's 260 million people.