This was conveyed by the minister during her meeting with families, who were "appreciative" of efforts made by the government, specially by Sawraj and Minister of State for External Affairs V K Singh, official sources said.
"The minister told them that bodies will be brought to India in a week's time and Singh will go to Iraq to oversee the entire process," a source said.
Singh is expected to fly out soon in a cargo plane to get bodies, sources said.
Davinder Singh, younger brother of Gobinder Singh, who was among those killed in Iraq, told PTI over phone after meeting Swaraj in New Delhi, "We were assured all kind of assistance, including the possibility of a government job to the next of kin."
On the demand of bringing the mortal remains of the 39 killed to India as soon as possible, the minister "assured us of all possible steps", he said.
The meeting with Swaraj lasted for about 45 minute, he said.
The family of the deceased sought financial help as their bread earners are dead and also government jobs for the next of kin, Kumar said.
"The minister told us that she would meet the CMs of the four states from where the Indians belonged to take a decision on possibility of providing a government job to the next of kin," he said.
Earlier in the day, the family members of those killed in Iraq met Punjab Congress chief and party MP Sunil Kumar Jakhar in New Delhi and he assured them "all possible assistance" to them.
Swaraj had told Parliament last week that all the 39 Indians, abducted by the ISIS in Iraq in 2014, had been killed and their bodies recovered.
As many as 40 Indians were abducted by the ISIS terror group from Mosul in Iraq but one of them escaped posing as a Muslim from Bangladesh, Swaraj had said in Rajya Sabha.
The remaining 39 Indians were taken to Badoosh and killed, she had said.
Search operations led to a mound in Badoosh where locals said some bodies were buried by the ISIS. Deep penetration radars were used to establish that the mound indeed was a mass grave, she had said, adding the Indian authorities requested their Iraqi counterpart to exhume the bodies.
Swaraj said the mass grave had exactly 39 bodies, with distinctive features such as long hair, non-Iraqi shoes and IDs. The bodies were then sent to Baghdad for DNA testing.
DNA testing by Martyrs Foundation established the identity of 38 Indians while there has been 70 per cent matching of the DNA for the 39th person, she had said.