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Danish police say the gunman believed to have attacked a free-speech debate and a Copenhagen synagogue was 22, born in Denmark and known to them because of past violence, the BBC reported.

The presumed gunman was shot and killed early on Sunday morning by police who were monitoring an address in the Norrebro district of the city.

The man's name has not been confirmed.

A film director and a synagogue guard were killed in separate attacks. Five police officers were also injured.

Police believe the gunman was acting alone.

He was known to them in connection with criminal gangs and had convictions for violent offences and dealing in weapons.

"It was the case that when the suspect was shot and killed during police action, he was armed with pistols," police commissioner Thorkild Fogde told a news conference.

Danish media have widely named him as Omar El-Hussein, but police did not release the suspect's identity.

The attacks began on Saturday, when the gunman fired shots at a cafe hosting a seminar on free speech. It was attended by cartoonist Lars Vilks, who has faced death threats over his caricatures of the Prophet Muhammad.

The gunman opened fire again in the early hours of Sunday outside a synagogue in Copenhagen.

The Danish intelligence service is investigating whether the gunman was copying the shootings in Paris last month, when 17 people were killed in attacks on the satirical magazine Charlie Hebdo and a kosher supermarket.