Police in the city of Heilbronn said no-one had been in the sports hall in nearby Wertheim at the time of the attack and it was not possible to enter the building because of fire damage.
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The attack came as a senior member of parliament in Chancellor Angela Merkel's conservative sister party, the Christian Social Union (CSU), spoke out in favour of tighter rules on granting asylum.
"Those who are not in danger should leave Germany as quickly as possible," Gerda Hasselfeldt told the Welt newspaper.
"We have to set clear priorities. We need our energy and resources for those who are fleeing war and persecution."
Read: Migrants missing in new boat sinking off Greece
The comments came as Hungary and Croatia traded threats as thousands of migrants poured over their borders, deepening the disarray in Europe over how to handle the crisis.
A proposal being argued over ahead of a meeting of European Union interior ministers on Tuesday would, if agreed, relocate 120,000 asylum seekers over the next two years around the whole bloc.
The number of migrants entering Germany, which recently reimposed border controls, was lower on Saturday, with 1,710 people registered, than on Friday, when the number was 1,985, police said.
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