Government said all the Indians in captivity were "unharmed" and it was "fully engaged" and "every possible effort" was being made to ensure their release.
As efforts were on to rescue all the Indians from the troubled areas in Iraq, Amnesty International claimed that hundreds of Indian nationals may be stranded in Najaf province.
The human rights watchdog claimed it had spoken over the phone with some Indian workers working for an infrastructure and construction company who said they were in danger as their employers had "refused to return their passports", thus rendering them unable to leave the Gulf country.
"Evidence has emerged which suggests that several hundred Indian nationals may be stranded in Najaf province of Iraq, unable to return home because their employer refuses to return their passports," Amnesty International India said in a statement.
Official sources here said the Indian mission in Baghdad has already contacted the company concerned and the matter is likely to be resolved soon.
"We have contacted the company concerned. The companies are already responsive. We are already working with them. They will have these people come across. There will be somebody from embassy who goes there, will sit down with the company and all the employees and decide on this," they said.
They also said India remained in contact with a number of countries in the region besides Iraqi authorities to resolve the crisis.
The sources further said government has indication about the identity of the captors and it was in touch with International Red Crescent in Iraq.
"All the 39 Indians in captivity are unharmed as of today," External Affairs Ministry Spokesperson Syed Akbaruddin said.
40 Indian construction workers were kidnapped from Mosul, Iraq's second largest city which was seized by Sunni militants, and one of them fled from the captors.