The Hamburg-based magazine cited a finance ministry document that predicted the annual costs would rise from about 16.1 billion euros (USD 18.2 billion) this year to 20.4 billion in 2020.
Much of the money would go toward basic benefits, housing support and language lessons for asylum-seekers, but the overall amount also includes spending on efforts to fight the reasons why people flee their home countries and seek refuge in Germany.
Finance ministry spokesman Juerg Weissgerber declined to comment on the figures but confirmed that federal officials were in discussion with representatives from Germany's 16 states about the cost of supporting refugees.
Heavy government spending on refugees has boosted Germany's economy in recent months, but has also stoked resentment among some Germans who believe that migrants are getting preferential treatment.
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A nationalist party, Alternative for Germany, has surged in recent polls and the number of attacks against refugee shelters has increased sharply over the past year.
Almost 1.1 million asylum-seekers entered Germany last year, although the government has stressed that the figure doesn't account for those who move on to other countries or return home.
It assumes that over half of those who are officially recognised as refugees will find work within five years.
Thanks to low unemployment and strong exports, Germany recorded a budget surplus of 12.1 billion euros last year on federal spending of 299.3 billion euros.
Finance Minister Wolfgang Schaeuble has stuck to his policy of balancing the country's budget despite the cost of dealing with the influx of migrants. Schaeuble said last week that Germany should be "able to manage the current major challenges without new debt.