He converted to Islam at the age of 14, but was radicalised by an extremist he worked with and organisations such as Call to Islam and Hizb ut Tahrir.
Mrs Dam told RNZ she had received four individual accounts corroborating Lukas' death.
"I need to believe (he is dead)," she said.
"I need peace and quiet and I need to get on. I don't know if he's alive. I don't know if he's in jail. I don't know if Islamic State have killed him."
She believed her son had been "manipulated".
"He has been abused and pushed into a fight that can't be won."
During his youth, Lukas was taken into the care of Copenhagen City Council.
Mrs Dam said she alerted social workers to her son's extremist beliefs and warned them of the possibility he might try to leave the country for the Middle East.
She has since received an apology from Copenhagen's deputy mayor Anna Mee Allerslev, who is responsible for integration and anti radicalisation programmes.
Officials at City Hall said its anti radicalisation programme was not given a report on Lukas until four months after he left.
Mrs Dam told RNZ she wanted to try to prevent the situation from happening again.
"I curse the hell out of Islamic State," she said.
Mr Key told RNZ he could not comment on the Dam case in particular, but said there was help for concerned New Zealand parents.
"In the first instance, if they reach out to any of the authorities - the police would be the first obvious port of call."
There had also been instances where the SIS had spoken with some individuals about their family members.
"If you work with family members, then that can actually be quite an effective way of trying to stop somebody joining with a group such as Isil."
