The notice comes just days after former Immigration Minister Michaelia Cash urged Dutton to refuse the 26-year-old American a visa on character grounds.
"People need to understand if you are going to commit domestic violence and then you want to travel around the world, there are going to be countries that say to you: `You cannot come in because you are not of the character we expect in Australia,'" Cash, who is now minister for women, told reporters last week.
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Australia isn't the first country to consider rejecting Brown. Britain, Canada and New Zealand have also refused to give him visas.
Brown was expected to arrive in Australia in December for a tour of Sydney, Melbourne, Brisbane and Perth. Activists have been campaigning to ban the singer from the country because of his brutal attack on his then-girlfriend, Rihanna, hours before the 2009 Grammy Awards. He pleaded guilty to felony assault and was freed from probation this year.
A publicist for Brown did not immediately respond to a request for comment.
The immigration department will decide whether to grant Brown a visa after the 28-day timeframe has expired.