Kalburgi, a rationalist and former vice chancellor of Kannada University, was shot dead at his home by an unidentified gunman on Sunday. Kalburgi was known for his progressive thoughts and had campaigned against idol worship.
"We will write to the CBI to take over the case. Till then, CID will investigate the case so that the evidence is collected," Law and Parliamentary Affairs Minister T B Jayachandra said.
Progressive writers expressed anguish over the killing of the 77 year old academic, who was an associated of rationalist Govind Pansare who was killed last year.
Karnataka police arrested a 25-year old right wing Bajrang dal activist Bhuvith Shetty, who tweeted that Bhagwan would be the next target.
"Then it was Anantmoorty and now its M M Kalburgi. Mock hinduism and die a dog's death. And dear K S Bhagwan you are next," Shetty tweeted in his handle @garudapurana on Sunday. He had deleted his handle after an uproar in the social media against the death threat.