The incident comes less than a week before the country goes to the polls in a national election in which the the populist FPO is tipped for significant gains.
According to reports, the FPÖ member, who stands as an independent local councillor in the southern state of Styria, is alleged to have raised his right arm and shouted “Heil Hitler” during a local council meeting.
The man has not been named and neither has the town where the incident is supposed to have happened.
Josef Riemer, an FPÖ chairman from the region, said in a statement that the party was taking the accusations very seriously.
He said the party had suspended the man’s membership until the case was resolved.
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The councillor has apparently denied the allegations and hired a lawyer.
It is illegal to make statements or own objects that glorify the Nazis in Austria, the birthplace of Adolf Hitler.
The FPÖ, which was actually founded by former Nazis but says it has left its past behind, has repeatedly thrown out officials in recent years over Nazi allegations.
Party chief Heinz-Christian Strache, who says anti-Semitism is a crime, has said Christian Europe shares an enemy with Israel in political Islam.
Austria’s anti-fascist Mauthausen Komitee, named after a Nazi concentration camp, published a brochure this year detailing 60 cases of Nazi-related incidents involving FPÖ members.
But Mr Strache is still expected to do well in Sunday’s national election and for the first time since 2000 his party has a good chance of joining the government in coalition with Sebastian Kurz’s ruling People’s Party.