The refugee agency UNHCR has recognized all 19 as refugees, but the United Arab Emirates authorities have told the group they must leave the country by April 11.
A 66-year-old woman and a four-year-old girl are among the 19. Three members of the group allege that Sri Lankan authorities had tortured them in 2011.
"For the UAE to return recognized Tamil refugees to a grave risk of torture in Sri Lanka would signal a total disregard for their well-being - and the most basic principle of international refugee and human rights law," said Bill Frelick of Human Rights Watch.
"Sri Lanka's treatment of Tamils they deem politically suspect is dismal and under no circumstances should the UAE deport this group there."
The 19, including six women, were part of a group of 46 Sri Lankan Tamil asylum seekers who fled Sri Lanka in October 2012, trying to reach Australia by boat.
After their boat ran into trouble, they were rescued by a Singaporean ship Oct 14. The 46 were taken to the port of Jebel Ali, south of Dubai, in the UAE.
In November, UNHCR recognized 39 of the 46 as refugees. Although UNHCR has been seeking third-country resettlement on their behalf, it has not yet been able to secure resettlement offers for 19 of them.
These are the refugees who are under threat of deportation, according to Sri Lankan sources, who told Human Rights Watch that UAE police have told the refugees that they must go back to Sri Lanka.