Sipping a $12 beer in one of the world’s wealthiest capitals, Andrei Medvedev reflected on the question hanging over him since he left the battlefields of Ukraine: Is he a hero or a war criminal?

He claims to have deserted from Russia’s notorious Wagner mercenary force during the monumental battle for the Ukrainian city of Bakhmut, and later to have escaped his native Russia by running across a frozen Arctic river. Now in Norway, Mr. Medvedev, 26, is seeking asylum, while providing information on Wagner to Norwegian authorities.

Since arriving in the country in January, Mr. Medvedev has voluntarily attended about a dozen interviews with Norwegian police officers investigating war crimes in Ukraine, including his potential role in them. Mr. Medvedev has described killing Ukrainians in combat and witnessing summary executions of comrades accused of cowardice. He claims that he did not participate or witness war crimes such as killings of prisoners of war and civilians.

“Yes, I have killed, I saw comrades die. It was war,” he said in an interview at an Oslo bar. “I have nothing to hide.”

His unlikely journey has made Mr. Medvedev one of only a handful of publicly known Russian combatants to seek protection in Europe after participating in the invasion. His asylum request is now forcing Norway to decide a case that pits the country’s humanitarian ethos against an increasingly assertive national security policy and solidarity with Ukraine.

To his lawyer, the credible threat of revenge facing Mr. Medvedev if he were sent back home qualifies him for asylum. And some Norwegian politicians have said that encouraging soldiers like Mr. Medvedev to defect would weaken Russia’s army and hasten the end of the war.

But as Norway evaluates his claim, it is facing pressure from activists in Ukraine and Western Europe, who say giving safe haven in Europe to Russian fighters, especially mercenaries like Mr. Medvedev, fails to hold Russians accountable for the invasion. And the former fighter may have complicated his own request with bar fights and detentions in Norway, and by briefly posting a video on YouTube suggesting he wanted to return to Russia.

More broadly, Mr. Medvedev’s case puts a spotlight on a policy dilemma that European governments have largely avoided grappling with in public: How should the region treat Russian deserters, and the hundreds of thousands of combatants in Russia’s war in Ukraine, in general?

“It goes to the core of who we are in Europe,” said Cecilie Hellestveit, an expert in armed conflict law affiliated with Norway’s human rights watchdog and a former member of the country’s asylum appeal board. “It forces us to re-evaluate our approach to human rights in a way that we have not been willing to do until now.”

The European Union, and affiliated states like Norway, have had to balance humanitarian needs with war crimes accountability before, most recently in processing immigration claims of people who fought in the Balkan and Syrian civil wars.

But the scale of the war in Ukraine, its proximity to the European Union, and the participation of two conventional armies means that the Russian invasion presents a much greater challenge to the region’s asylum system, Ms. Hellestveit said.

Four months after Mr. Medvedev requested asylum, his claim remains pending. Norway’s immigration agency said all asylum cases filed by Russians who fled to evade military service were on hold while they analyze the human rights conditions in the country. The agency said it could not comment on individual applications for privacy reasons.

Some humanitarian law experts in Norway say Mr. Medvedev’s unresolved request reflects the government’s reluctance to bring further attention to a case that could divide the public, get ahead of the policies of other European states and strain relations with Kyiv. Norway has been a fervent supporter of the Ukrainian cause, committing $7.5 billion worth of economic and military aid, and accepting about 40,000 Ukrainian refugees.

“This case has a lot of conflicting rights, a lot of conflicting obligations and a lot of conflicting politics,” said Paal Nesse, the head of Norwegian Organization for Asylum Seekers, a nonprofit providing legal aid to applicants.

Norway and E.U. countries have struggled to formulate a common approach for asylum claims submitted by Russians who have fled the country to avoid military service, a much larger group of applicants than men who engaged in combat, like Mr. Medvedev.

The European Union’s Agency for Asylum said in a written response to questions that it is up to member states to decide who deserves protection.

Pavel Filatiev, a former Russian paratrooper who requested asylum in France after fighting in Ukraine, said he was waiting for a decision eight months after submitting his application. A third publicly known Russian deserter in Europe, a former army mechanic named Nikita Chibrin, has had a pending asylum claim in Spain since November.

The legal uncertainty, financial problems and social isolation are difficult to bear, Mr. Filatiev said in a phone interview, but he added that he considered himself fortunate and was grateful to his French hosts.

“I understand that my decision to leave will always haunt me,” he said.

Mr. Medvedev’s has a troubled history of antisocial behavior. Already, he has been detained twice in Norway for getting into fights in bars and once in Sweden for entering the country illegally. (He was returned to Norway.) In Russia, he spent four years in jail for robbery and getting into fights, according to court records.

People who know him have said those actions could be a consequence of a lifetime of trauma: in a violent family home, a Siberian orphanage and Russian jails, and on Ukrainian battlefields.

In addition to his run-ins with the law, Mr. Medvedev said he had also repeatedly clashed in Oslo with Ukrainians, most recently while visiting a local Soviet military memorial on Victory Day.

Such run-ins have underlined the tensions between the Russian defectors and Ukrainian refugees across Europe. Natalia Lutsyk, the head of the Ukrainian Association in Norway, said the lack of international cooperation prevented Norway and other nations from thoroughly investigating war crimes committed in Ukraine.

“Thus, Medvedev and his companions remain unpunished,” she added.

The New York Times spent several weeks interviewing Mr. Medvedev and researching his personal history since he left the front in November and went into hiding in Russia. His account of his military service has contained contradictory or unverifiable claims. Some basic facts of his life, however, have been corroborated by public records and interviews with acquaintances.

The weight of this evidence shows that Mr. Medvedev enlisted with Wagner in July 2022, two days after finishing his latest prison sentence.

Wagner’s founder, Yevgeny V. Prigozhin, in April called Mr. Medvedev a “jackass who spent two days in Wagner, who can’t identify anyone.” After his escape to Norway, Mr. Prigozhin called him dangerous. He has not publicly threatened Mr. Medvedev.

In an interview in Oslo, Mr. Medvedev described his new living conditions, provided mostly by the Norwegian state. According to him, they include a house, home visits by a Norwegian language teacher, an integration assistant, ski and mountain bike trips, and “Taco Saturdays” with a personal security detail.

He also claims to be a subject of a bidding war between filmmakers, an assertion that could not be verified.

But days after the interview, Mr. Medvedev declared that he had contacted the Russian Embassy to get help returning home.

“I hope that I could find peace and calm here, that I could leave behind the politics, the war, the army,” he said in a video published on YouTube. “It was not to be.”

He later deleted the videos and declined to speak again when contacted by phone.

His lawyer, Brynjulf Risnes, said his public comments should not influence the asylum claim, which is decided on humanitarian grounds. But Mr. Medvedev’s violent past and controversial behavior, which has turned him into a minor local celebrity, have confused and alienated many Norwegians, sapping sympathy for Russian defectors.

Under Norwegian law, refusing to fight in an illegal war may grant a right to asylum. However, this right does not apply to war criminals, and local prosecutors can charge people who they believe have committed war crimes elsewhere.

A Norwegian criminal police spokesman said Mr. Medvedev was a witness, not a suspect, in its investigation of war crimes in Ukraine, and that, to date, officers “have not found grounds for charges.”

Mr. Medvedev said his cooperation had helped investigators locate Wagner facilities in Ukraine and Russia and map the group’s structure.

The case is also being followed by Ukrainian officials, who are conducting their own investigation of Mr. Medvedev. Shortly after his arrival in Norway, Ukraine’s ambassador in Oslo told local news media that her government could request his extradition.

Such a request would present Norway with another dilemma, forcing it to choose between a show of support for an ally and upholding the basic principle of its asylum law. This law states that an asylum seeker cannot be sent to a country where they may not get a fair trial.

The office of Ukraine’s prosecutor general said in a written response to questions that it checked all Russian servicemen who arrive in foreign countries for potential participation in war crimes, and that it had requested Norway’s legal assistance in investigating Mr. Medvedev.

Mr. Medvedev said he had refused to see Ukrainian investigators who asked to meet to him in Norway.

“They are always after me,” he said. “I’m helping them to end this war.”

Constant Méheut contributed reporting from Paris, Alina Lobzina from London and Natalia Yermak from Kyiv, Ukraine.