According to the Falter weekly, Udo Landbauer of the Freedom Party (FPÖ), which is in the national governing coalition, is deputy chair of a student fraternity behind a songbook containing the texts.
The left-leaning magazine reported the lyrics of one song reads: "In their midst comes the Jew Ben Gurion: 'Step on the gas, old Germanics, we can make it to seven million'."
Six million Jews were killed in the Holocaust during World War II, many of them in gas chambers in Nazi death camps. David Ben-Gurion was the first prime minister of Israel.
Other songs pay tribute to the Condor Legion, the Nazi unit responsible for the bombing of Guernica in the Spanish Civil War, as well as paratroopers behind atrocities in Crete in World War II, Falter added.
The allegations came ahead of an election in Lower Austria on Sunday. Opinion polls predict strong gains for the FPÖ, putting them in third place -- as in national elections last October -- on between 16 and 21 percent.
Landbauer said he was "shocked" by the texts, which he said he had only become aware of on Tuesday, and that he was freezing his membership of the fraternity.
"When this book was printed (in 1997) I was 11 years old," he said in a statement, adding that "neither I or the FPÖ have anything to do with anti-Semitism, xenophobia or totalitarianism".
Calls to quit
Bernhard Ebner, the head of the centre-right People's Party (ÖVP) in Lower Austria, said the allegations were "unbelievably serious" and needed to be "completely cleared up".
The centre-left Social Democrats (SPÖ) said Landbauer should resign immediately, a call echoed by the liberal NEOS party if the claims turned out to be accurate.
Sunday's election is the first since the FPÖ became junior coalition partners to the ÖVP in December under Chancellor Sebastian Kurz, obtaining the interior, foreign and defence ministries.
Despite its leadership saying it rejects extremism and its leader Heinz-Christian Strache visiting Israel, Austria's main Jewish organisation the IKG earlier this month said it would refuse to meet FPÖ ministers.
The IKG said on Tuesday that it would boycott a commemorative event on Thursday in Austria's temporary parliament ahead of International Holocaust Remembrance Day on Saturday.
The party was formed by former Nazis in the 1950s and Strache, 48, now vice-chancellor, dallied with neo-Nazism in his youth although he now says this was when he was "stupid, young and naive".
Kurz of the ÖVP, who has rejected criticism of his coalition with the FPÖ by saying the party should be judged on its actions not its past, on Tuesday took to Twitter to call the song texts "racist, anti-Semitic and absolutely sickening".
"It needs to be fully cleared up and those responsible brought to account," Kurz wrote.
Landbauer, whose mother is Iranian, in November called his ÖVP rival in the election, former interior minister Johanna Mikl-Leitner, "Muslim-mummy-Mikl" because of her support for "multi-cultural madness".