Official flag rules in Denmark state that, unless special authorisation is given, it is not permitted to fly any other national flags than the Danish Dannebrog, a white Scandinavian cross on a red background.
Police visited the family at their home on Monday and asked them to lower the American flag, reports local media Jydske Vestkysten.
The flag has flown from a flagpole in the Hedegård family's garden for the last month. Police received reports of a prohibited flag being flown at the address, according to the report.
The family was reportedly advised it could receive a fine of 2,500 kroner ($375) if it did not lower the flag.
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Rikke Hedegård told Jydske Vestkysten that she was disappointed that nobody spoke to her before reporting the flag.
“My thoughts are that if someone was so offended, why didn't they stop by and mention it? We go outside almost every day, so just stop by and listen to our story,” she said.
The family chose to fly the flag because they like American culture, Hedegård said.
“We feel we are a part of American culture in Denmark,” she told Jydske Vestkysten.
“I could understand if it was a Nazi or Isis flag, but an American flag, I don't understand that at all. But it's probably because [complainants] are not part of the culture,” she said.
There are some exceptions to Danish flag rules – the Greenland and Faroe Islands flags, along with flags of the other Nordic countries, the EU and the UN may be flown, according to Ministry of Justice guidelines.
If permission is given to fly other countries' flags, this is usually on condition that a Danish flag of at least equal size be hoisted alongside the foreign flag.
The ban on flying foreign flags in Denmark dates back to 1915.