Police are investigating whether a couple who killed four people at a kosher grocery store in the US state of New Jersey were motivated by anti-Semitic beliefs, after the local mayor labelled the attack a "hate crime".

Key points: Police are investigating links between the shooters and an anti-Semitic Black Israelite group

Police are investigating links between the shooters and an anti-Semitic Black Israelite group The pair killed four people before dying in a shootout with police

The pair killed four people before dying in a shootout with police The attack has been labelled a "hate crime" by Jersey City mayor Steven Fulop

Three people were killed inside the store after a man and woman pulled up in a rental van and began firing at the Jersey City store yesterday.

The two attackers, identified as David N Anderson, 47, and Francine Graham, 50, were killed in the ensuing firefight with police, along with one police officer.

Investigators believe Anderson and Graham had previously identified as Black Hebrew Israelites, a movement whose members have been known to rail against white people and Jews, according to a law enforcement official who could not be named publicly.

David Anderson and Francine Graham were both killed by police. ( AP: Seth Wenig )

Authorities found social media postings from at least one of the killers that were anti-police and anti-Jewish, the official said.

The FBI on Wednesday searched the Harlem headquarters of the Israelite Church of God in Jesus Christ, the formal name of the Black Hebrew Israelite group, according to the official.

Some branches of that church have been named as hate groups by the Southern Poverty Law Centre, which tracks race-based extremism in the US.

While authorities are yet to publicly confirm a motive for the attack, several political figures have drawn links between the attack and an anti-Semitic ideology.

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Jersey City Mayor Steven Fulop said surveillance video of the attackers made clear they had committed a "hate crime" against Jewish people.

Mr Fulop said it showed the killers deliberately made their way toward the kosher market, passing many other possible targets along the way, before calmly and promptly opening fire.

"The video is clear that the shooters bypassed potential targets near the store," he tweeted.

"It is also clear the shooters have indicated on social media favourable sentiment towards groups that show anti-Semitism."

New York Mayor Bill de Blasio said the attack was a "premeditated, violent, anti-Semitic hate crime," while New York Governor Andrew Cuomo called it a "deliberate attack on the Jewish community."

They announced tighter police protection of synagogues and other Jewish establishments in New York as a precaution.

While examining the influence the couple's beliefs had on the attack, US state and federal law enforcement officials warned they had not firmly established a motive.

"The why and the ideology and the motivation — that's what we're investigating," New Jersey Attorney General Gurbir Grewal said, adding that authorities were also trying to determine if anyone else was involved.

The Jewish community is demanding answers as to whether the attack was targeted. ( AP: Mark Lennihan )

Anderson and Graham were also prime suspects in the murder of a taxi driver found dead in car boot in nearby Bayonne over the weekend, Mr Grewal said.

Anderson served about four months in prison in New Jersey on weapons charges and was paroled in 2011, authorities said.

Two of the victims at the store were identified by members of the Orthodox Jewish community as 31-year-old Mindel Ferencz, who owned the grocery with her husband, and 24-year-old Moshe Deutsch, a rabbinical student from Brooklyn.

Authorities identified the third victim as Miguel Douglas, 49.

"The report from the Jersey City mayor saying it was a targeted attack makes us incredibly concerned in the Jewish community," Evan Bernstein, regional director of the Anti-Defamation League, said.

"They want answers. They demand answers. If this was truly a targeted killing of Jews, then we need to know that right away, and there needs to be the pushing back on this at the highest levels possible."

ABC/AP