Some 300 to 400 Torah scrolls and thousands of vestments, crowns and other articles are being held in museums and storage in Russia, said Slomo Koves, executive rabbi of the Unified Hungarian Jewish Congregation.
Seized by Nazi Germany during the war and then by the Red Army in 1945, many of the items had been stored for safety in the Hungarian National Bank during the Holocaust.
Up to 600,000 Jews died at the hands of the Nazis and their allies, according to Budapest's Holocaust Memorial Center, and Hungary's Jewish community, now numbering around 100,000, still faces anti-Semitism.
Last week, a far-right politician urged the government to draw up list of Jews who posed a "national security risk."
In June, Nobel Peace Prize laureate Elie Wiesel repudiated a Hungarian state award he received after an official from the Hungarian government participated in a ceremony honoring a writer who was a member of Hungary's WWII far-right parliament.
Koves said he had sent a letter to Russian authorities about the religious artifacts, which are part of a larger body of art and valuables that Hungarian authorities say the Soviets took from the country when the war ended.
"Our request is that if we're not speaking about looted art, at least we can speak about all the pieces that are important to the Jewish community," Koves said.
He had asked for Moscow to allow a delegation to document the items and then work out a process for their return.
Hungarian Foreign Minister Janosz Martonyi has also written to Moscow to say the issue had been "unfortunately dragging on for a long time now" and saying it was "of great national importance for the whole country".
Russia's Foreign Ministry declined immediate comment.
About one tenth of the six million Jews who were killed during the Holocaust came from Hungary. Most were deported to death camps after Nazi Germany occupied the country in 1944, often after soldiers seized their possessions.
Historians have documented stories of people who risked their lives or died trying to protect Torah scrolls. Some who ran into burning synagogues or hid them on their person when they were deported to death camps.
Scrolls in museums
Torah scrolls are at the heart of Jewish worship, and a community needs to obtain them before it builds a synagogue.
Painstakingly copied by hand on parchment, they contain the five books of Moses and are kept in an arch of a synagogue, usually facing east towards Jerusalem. Koves said the ones in question were hundreds of years old and "priceless".
According to a catalogue compiled by the Conference on Jewish Material Claims against Germany, 344 Torah scrolls were turned over by Russia's Special Archive, now part of the Russian State Military Archive, to the State Historical Museum.
The Hungarian group said Russia also holds Torahs in the Nizhny Novgorod Museum. Some scholars say these originated in Hungary while others say they came from Russian synagogues.
"When we speak of Torah scrolls, there is no question that they belong to the Jewish community, and we are all heirs of the survivors," Koves said.
"Today, when we don't have to endanger our lives but just have to bring it to people's attention, this is our first duty as Hungarian Jews." Some 8,000 Jewish Holocaust survivors live in Hungary.