The 46-year-old Bosnian, described by Swedish media as "an Islamic State jihadist", arrived in the country on Monday with his four children.
The Sydsvenskan newspaper, which broke the story, said he had been flagged as a suspect in the Schengen Information System (SIS) -- Europe's system for sharing national security data -- at France's request.
He had flown into Malmo airport in southern Sweden from the Bosnian city of Tuzla carrying a recently-issued passport in his own name.
The man then demanded asylum and, despite being a suspected jihadist, was transferred to an immigration detention centre as Sweden considers the Geneva Convention on refugees to take precedence over police matters.
"Any individual is entitled to do this," Migration Office spokesman Fredrik Bengtsson told AFP. "This does not prejudge the decision that will be taken after his case is examined."
Police criticised the decision to hand the man over to immigration officials, saying he should have been deported.
"As soon as these people play their wildcard and say 'Asylum!', the gates of paradise open for them," Sydsvenskan quoted border police chief Leif Fransson as saying.
Sydsvenskan reported that in 2010 the man had been arrested while living in Bosnia after a search of his home found arms and jihadist propaganda.
He had also been arrested in Turkey in February, suspected of having fought for IS, the regional newspaper added. He was deported back to Bosnia where police had seized his passport and banned him from leaving the country.
As a Bosnian national he has an extremely slim chance of winning asylum as his country, a candidate for EU membership, is considered safe by many of the bloc's members.
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