The warning by Germany's Federal Criminal Police Office (BKA) comes after a mayoral candidate in the eastern city of Cologne was stabbed in the neck in an attack apparently motivated by her work with refugees.
The BKA said there were 285 attacks on migrant centres across Germany during July, August and September -- more than in the whole of 2014, when only 198 offences were recorded.
There have been 576 offences targeting refugee hostels since the start of the year, including cases of criminal damage and inciting hatred, 523 of which were committed by far-right extremists, the BKA said.
Independent candidate Henriette Reker became the most prominent victim of a growing backlash against a huge influx of migrants to Germany on Saturday when a man attacked her with knives as she campaigned in the Cologne vote.
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She survived serious injuries and went on to win the election.
More than 630,000 people fleeing war and misery in the Middle East and Africa have landed on Europe's shores so far this year.
German security experts are bracing for a rise in unrest, with domestic intelligence chief Hans-Georg Maassen warning in late September of a radicalisation of right-wing groups and "a greater willingness to use violence" by all extremist groups.