Yemeni authorities have blamed IS for the Friday attack on the refuge for the elderly operated by Mother Teresa's Missionaries of Charity in main southern city Aden.
"According to our information, the extremists who attacked the elderly care home in Aden have kidnapped priest Tom Uzhunnalil, a 56-year-old Indian, who was taken to an unknown location," a Yemeni security official told AFP.
16 dead in gun attack on Yemen care home
"We are aware that no group has yet claimed the criminal attack... but information points to the involvement of Daesh," said the source, who asked to remain anonymous, using an Arabic acronym for IS.
Gunmen stormed the refuge killing a Yemeni guard before tying up and shooting 15 other employees, officials said.
Four foreign nuns working as nurses were among those killed.
The Vatican missionary news agency Fides identified the nuns as two Rwandans, a Kenyan and an Indian, adding that the mother superior managed to hide and survive while an Indian priest was missing.
India's Foreign Minister Sushma Swaraj on Sunday tweeted: "Father Tom Uzhunnalil -- an Indian national has been abducted by terrorists in Yemen."
"Yemen is a conflict zone. We do not have (an) embassy there. But we will spare no efforts to rescue Father Tom Uzhunnalil," she wrote.
The internationally recognised government in war-torn Yemen is grappling with both an Iran-backed rebellion and a growing extremists’ presence.
The Vatican's Secretary of State Pietro Parolin has said Pope Francis "was shocked and profoundly saddened" to learn of "this act of senseless and diabolical violence."
Al-Qaeda and IS have stepped up attacks in Aden, targeting mainly loyalists and members of a Saudi-led coalition battling Huthi rebels and their allies since March last year.
Al-Qaeda distanced itself from the mass shooting Friday, saying it was not responsible.
President Abedrabbo Mansour Hadi has declared Aden to be Yemen's temporary capital as Sanaa has been in the hands of rebels since September 2014.
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