The boys, who have not been named, say the woman encouraged them to film her having sex with them, newspaper Eskilstuna Kuriren reports.
She then urged them to watch the films and to phone her when they missed her, they said.
Eskilstuna Kuriren was given access to the films and believes they show the acts described by the boys, and that the woman can clearly be identified. She rejects the accusations.
The boys told Eskilstuna Kuriren they tried to inform social services and the police about what was happening but nobody listened to them.
They then went to the newspaper with their story on the advice of relatives living in another part of Sweden.
“The boss at the home forced us into it and exploited us for sex. She knew we had to, and that nobody would help us,” one said.
The boys say they had sex with the woman on four or five occasions, at a hotel and in her own home in the Sörmland region.
On one occasion all three were in bed together, they say, but mostly one of the boys had sex with the woman while the other filmed.
The woman also offered the boys alcohol, they claim, saying it would help them enjoy the experience and despite them being under 18, the legal age for drinking alcohol.
They allege that she promised to buy them clothes and toiletries if they had sex with her, but she also told them she would destroy important documents and get the migration agency to deport them if they turned her down.
They also faced sexual advances from other people in the woman’s social circle, they say.
In an interview with Eskilstuna Kuriren one of the boys says they would be “stoned to death” if they ever went back to Afghanistan and their story got out.
The boys were recently moved to a home in a different municipality. Local authorities there contacted the police after learning of their allegations.
Police confirmed to the newspaper that an investigation into sexual offences was ongoing. They would not specify what the alleged offences were.
The boys say they arrived in Sweden in November after they were urged by their families to flee Afghanistan and seek a better life.