The issue has been discussed several times before, with some municipalities forcefully opposed to such a move, while others who do house and school the children say it is better they live in towns where they are welcome.
Sweden welcomes more underage asylum seekers than any other EU countries. Last year, 2,657 cases were lodged. Most come from war-torn Afghanistan, where Sweden is part of the Nato-led Isaf mission, and Somalia.
A lack of residences mean the children and teenagers are stuck in temporary lodgings, notes Billström, while their asylum cases are tried by the Migration Board.
"The number of municipalities doing their share has increased but not enough to provide enough housing," Billström says.
It is unclear whether Billström's initiative will be supported by a majority in parliament.
There is also strong opposition to such a move from many municipalities.
"From a psychological and pedagogical perspective, I don't think it's much fun to be sent to a municipality where you are not welcomed," says Per Åhden, who works in Skellefteå in northern Sweden, known for taking care of large numbers of lone-comers.
Several local politicians think there needs to be more clarity about how much financial compensation they can demand from the central government to provide housing and education for the young migrants.
Katrineholm municipality, two hours from the capital Stockholm, has not signed up to take care of the children.
"We can't take the full responsibility," say local politician Göran Dahlström of the Social Democrat party.
"The government can't legislate and then not assign enough resources," he says.