The film, a no-holds-barred look at how a corrupt local mayor crushes all who oppose him to arrive at his goals at all cost, has even prompted a Russian Orthodox activist to call for it not be screened in Russia.
A portrait of Putin that is often seen looking down on the mayor creates what many see as a link with the Kremlin and the Russian leader's governing style.
Putin critics say the story mirrors life in Russia in the 15 years since the former KGB spy first rose to power, with corrupt state officials enriching themselves and enjoying impunity.
Russia's Culture Ministry co-financed the film but now says it blackens Russia's image just to win international acclaim.
"Films focused not only on criticism of current authorities but openly spitting on them..., filled with a sense of despair and hopelessness over our existence, should not be financed with taxpayers' money," Culture Minister Vladimir Medinsky said in a newspaper interview this week when asked whether the ministry would support similar films in the future.
Though Leviathan premiered in mid-2014, cinemas in Russia will start screening it only in February, with foul language muted to comply with Russian profanity laws.
The film, largely shot in the village of Teriberka on the Barents Sea in Russia's far north, has already won a dozen awards abroad, including a Golden Globe.
On Thursday, it captured an Academy Award nomination in the Foreign Language Film category along with four others.
The authors say their film was partly inspired by a story from the United States. Many in the director's home country, however, see it as aimed directly at Putin's Russia, though Zvyagintsev himself has sought to steer away from such links.
"It did not matter in what setting the events of this drama unfolded. The story of conflict between the individual and the authorities is universal," Zvyagintsev's website says.
The director himself was not immediately available to comment.
Medinsky began complaining about the film last year when Leviathan received favourable reviews at the Cannes film festival, one of cinema's most prestigious events. He said he did not like the film's excessive profanity.
In an interview published on Thursday he complained to Izvestia newspaper, which is sympathetic to the Kremlin, that Leviathan had no positive characters. He said the story was not specific to Russia and could have been played out anywhere.
"I hope in the future Andrei Zvyagintsev, a very gifted man, will make a film with the assistance of the Culture Ministry that will not feature this existential hopelessness," he said.
"A film that will make one want to get up, get out on the street and do something good, right, without delays - right here and right now. You don't get that after 'Leviathan'."
Izvestia also quoted a Russian Orthodox activist calling for the film, which shows dubious cooperation between the mayor and local clergy, not to be shown in Russia because it vilifies the Russian Orthodox Church.
Putin, whose popularity soared over the annexation of Crimea in 2014 but could be threatened by an economic crisis, has not commented publicly on the controversy and few Russians have yet seen the film.
The success of Leviathan abroad gets limited coverage in mainstream Russian media, most of which are loyal to the Kremlin.
Zvyaginstev told the independent broadcaster Dozhd he was bewildered by the treatment he and his film were getting from state TV and that he felt himself being "isolated".
He said no officials congratulated him on the Golden Globe award for Best Foreign Language Film, the first such award for Russia after the 1969 triumph of War and Peace produced in the Soviet Union.
The last time a Russian film won an Oscar was in 1994 with Burnt by the Sun, an intimate study of a family destroyed by Stalinist purges in the 1930s by director Nikita Mikhalkov, now an outspoken supporter of Putin.
Winners of the 87th Academy Awards will be revealed in Los Angeles on Feb. 22. (Editing by Timothy Heritage and Peter Millership)