GERMERING, Germany— Valmir Sahiti, 25 years old, has spent the last two weeks living in a nursing home in the long-shot hope of a new life.

Mr. Sahiti, an asylum-seeker from Kosovo, is staying with five relatives in a makeshift refugee center set up on several turquoise-carpeted empty floors of the home. He says he is aware the German authorities may soon send him back, but he insists the chance to live in Germany is worth the attempt.

“We love Germany,” Mr. Sahiti, a Kosovar Albanian, says. “Albanians have a saying: If God wants it, he will do it.”

A sudden crush of migrants from Kosovo is straining Germany’s capacity, forcing officials in especially hard-hit Bavaria to house people such as Mr. Sahiti in emergency shelters, from the Germering nursing home to public gyms to the VIP spectator area of Munich’s Olympic Stadium. The influx is feeding divisive national debates over immigration, asylum policy and European integration.

Officials say Kosovars have virtually no chance of being granted asylum because they aren’t fleeing war or persecution. Nevertheless, a surge of close to 20,000 people has arrived from the poor Balkan country by bus, rail, and taxi since around the start of the year, overwhelming government officials already trying to house rising numbers of refugees from the Middle East and Africa.

Officials struggle to explain the sudden increase, but point to a combination of factors: The draw of Germany’s relatively strong economy and generous welfare benefits, economic desperation in Kosovo, an easing of border crossing rules between Kosovo and longtime foe Serbia, and rumors on social media that it has gotten easier to get asylum.

Kosovo, which was part of Yugoslavia during the communist era and later fought for independence from Serbia, was recognized as an independent nation by the U.S. and most of the EU in 2008. But it suffers from one of the weakest economies in Europe and an unemployment rate of 35%.

Officials are working to dispel the notion that Germany is a place where Kosovars can walk into plum jobs and government benefits. They have pledged to accelerate asylum review for Kosovars with the goal of finishing the process within weeks rather than months. And, showing they are taking action, officials announced Tuesday they had deported 30 Kosovars via charter flight.

Last week, a cabinet minister in Bavaria’s state government flew to Pristina, the capital of Kosovo, and went on national television to tell viewers they should stay home.

“Everyone in Kosovo must know,” the minister, Beate Merk, said, “asylum law provides no way into Germany.”

The government’s efforts point to the political sensitivities involved. Even before the recent surge, politicians on the right said the government wasn’t doing enough to prevent abuses of the asylum system, which provides some welfare benefits to applicants.

More broadly, the rise in migrants highlights worries in the German public about the costs of European integration, from eurozone bailouts to refugees coming in via other EU countries.

“We cannot let ourselves be exploited,” says Christian Bernreiter, head of the district government of Deggendorf in rural Bavaria. “A lot is coming together right now—the issue of Greece is worrying people, and the issue of refugees is worrying people.”

Mr. Bernreiter’s district hosts a 501-person home for asylum seekers that filled up quickly after it opened in early January. In a single 36-hour period Feb. 4 to 6, 300 asylum seekers from Kosovo arrived in Deggendorf, he said, forcing two emergency shelters to be set up in nearby districts.

Germany reported 202,834 people applying for asylum last year, a 60% increase from 2013 and the highest number in Europe. While Germany had been gearing up for more refugees from war-torn regions, the sudden jump in Kosovars caught officials by surprise.

Last year’s asylum seekers came mainly from Syria, Serbia, Eritrea, Afghanistan and Iran. This year, the German government says around 18,000 Kosovars began the asylum-seeking process between Jan. 1 and Feb. 12, more than double the number of their countrymen who applied for asylum all of last year. For the Kosovars whose asylum applications were resolved in January, only 0.3% got a favorable decision.

Frontex, the European Union’s border policing agency, says it detected almost 12,000 migrants, predominantly Kosovars, crossing the border from Serbia to EU member Hungary in January and roughly another 11,000 in the first half of February. The EU’s European Asylum Support Office says just about all Kosovar applicants have been seeking asylum in Hungary, Austria or Germany.

Mr. Sahiti, who accompanied his uncle, 40-year-old Jetuliah Sahiti, and the uncle’s wife and three children, said he made no more than $230 a month on odd jobs such as installing windows and had to provide for his pregnant wife, daughter and parents. Both men had lived in Germany as refugees during the war and still speak broken German.

The younger Mr. Sahiti said he and his uncle had recently heard that the way to Germany from Kosovo had become easier and less expensive and pooled their savings to pay some $2,800 for the trip. He left his own family behind—he hopes to bring them to Germany later or work for a few years and return—and joined his uncle’s family on the trip.

The six took a bus from Pristina to the Serbian capital of Belgrade, then another to a town near the border with Hungary. They walked for four hours, at times through forest, to a Hungarian village and took a taxi on a 10-hour drive to Munich, where the group turned themselves in to the police and asked for asylum.

The Sahitis received clothes at the asylum center they were first sent to in Munich and will be fed and sheltered by the state until the final decision is made. Gazi Durak, a supervisor of their shelter, said the cases of newly arrived Kosovar refugees likely wouldn’t be considered until April.

“We just wanted to come to Germany,” Mr. Sahiti said. “If we get asylum, good. If we get sent home, there’s nothing we can do.”