The unidentified gunmen launched simultaneous attacks in Dagestan's largest city of Makhachkala and in the coastal city of Derbent.

Russia's Investigative Committee said it had opened criminal probes over "acts of terror" in Dagestan, a largely Muslim region of Russia neighbouring Chechnya.

The leader of Dagestan, Sergei Melikov, wrote on Telegram: "This evening in Derbent and Makhachkala unknown (attackers) made attempts to destabilise the situation in society."

"We know who is behind these terrorist attacks and what objective they are pursuing," he added later, without specifying but referring to the war in Ukraine.

"We must understand that war comes to our homes too. We felt it, but today we face it," he said.

Melikov said the "active phase" of operations in Derbent and Makhachkala had ended and that "six bandits have been liquidated".

The authorities will try to find "all the members of these sleeper cells who prepared (the attacks) and who were prepared, including abroad", he added.

Russian officials said police had killed four gunmen in Makhachkala and two in Derbent.

Patriarch Kirill, head of the Russian Orthodox Church and a fervent supporter of the Kremlin, said the "enemy" was seeking to destroy "inter-religious peace" in Russia, without naming who was responsible.

The attacks targeted "two Orthodox churches, a synagogue and a police checkpoint", the National Anti-terrorism Committee said in a statement to RIA Novosti news agency.

"As a result of the terrorist attacks, according to preliminary information, a priest from the Russian Orthodox Church and police officers were killed."

The Russian Orthodox Church said archpriest Nikolai Kotelnikov was "brutally killed" in Derbent.

In all, six officers were killed and 12 wounded in the attacks, the spokeswoman for Dagestan's interior ministry, Gayana Gariyeva, told RIA Novosti.

The ministry later added that a local police chief had died from his wounds.

Russia's National Guard meanwhile said one of its officers had been killed in Derbent and several others wounded.

The Dagestan interior ministry said a total of 16 people, including 13 police officers, were wounded in the attacks.

In a separate incident, the ministry said gunmen had also shot at a police car in the village of Sergokal, 65 kilometers from Makhachkala, wounding one officer.

Sunday is a religious holiday, Pentecost Sunday, in the Russian Orthodox Church.

Dagestan's interior ministry said 19 people took shelter inside a church in Makhachkala and were later led out to safety.

Gunmen also attacked synagogues in both cities.

"The synagogue in Derbent is on fire," the chairman of the public council of Russia's Federation of Jewish Communities, Boruch Gorin, wrote on Telegram.

"The synagogue in Makhachkala has also been set on fire and burnt down," he said.

The rabbi of Makhachkala, Rami Davidov, later told RIA Novosti that no one was killed or injured there.

The Russian Jewish Congress said on its website the Derbent synagogue was attacked about 40 minutes before evening prayers.

Gunmen fired at police and security guards and threw in Molotov cocktails, it said, adding that the attack in Makhachkala was similar.

State news agency TASS cited a law enforcement source as saying the "gunmen who carried out attacks in Makhachkala and Derbent are supporters of an international terrorist organisation", without naming it.

Russia's FSB security service in April said it had arrested four people in Dagestan on suspicion of plotting the deadly attack on Moscow's Crocus City Hall concert venue in March, which was claimed by the Islamic State group.

Militants from Dagestan are known to have travelled to join IS in Syria, and in 2015, the group declared it had established a "franchise" in the North Caucasus.

Dagestan lies east of Chechnya, where Russian authorities battled separatists in two brutal wars, first in 1994-1996 and then in 1999-2000.

Since the defeat of Chechen insurgents, Russian authorities have been locked in a simmering conflict with Islamist militants from across the North Caucasus that has killed scores of civilians and police.