The body of Hemant Kumar Lohia, 57, the region's director general of prisons, was found at his home on Monday night in the Jammu region, police said.
Police said a household helper was the main suspect but an Islamist militant group said it had targeted and killed Lohia.
Muslim-majority Kashmir is divided between mostly Hindu India and Muslim Pakistan which both claim it in full.
Separatist Muslim groups have fought against Indian security forces in its part of Kashmir since the late 1980s.
Senior police officer Mukesh Singh said Lohia's throat had been cut and his body bore burns. The initial investigation suggested it was not a "terror act" but police were investigating, he said.
The People's Anti-Fascist Front (PAFF), a militant group that emerged after India's government reorganised its only Muslim-majority state into two federally administered territories in 2019, said it had assassinated Lohia.
Police have blamed groups like the PAFF for targeted killing but militants have not killed any security official of Lohia's seniority in recent years.
"This is just a beginning of such high profile operations," the PAFF said in a statement on social media, adding that the killing were a "small gift" to Home Minister Amit Shah, who arrived in Kashmir on Monday on a three-day visit.
Reuters could not immediately verify the authenticity of the PAFF statement.
(Writing by Devjyot Ghoshal; Editing by Robert Birsel)
By Fayaz Bukhari