Mr Mevani's associate Kaushik Parmar, who received the threat on a mobile phone this afternoon, lodged a complaint in Vadgam, Mr Mevani's assembly constituency in Banaskantha district.
"Parmar handles Mevani's office in Vadgam. Based on his complaint, we registered an FIR against the caller who had identified himself as Rajvir Mishra," said police sub-inspector RP Zala.
The man allegedly called Mr Parmar and threatened to shoot Mr Mevani, Mr Zala said.
A case under IPC section 507 (criminal intimidation by anonymous communication) was registered.
Mr Mevani took to Twitter after learning about the threat, giving out details such as the number from which Mr Parmar received the threat call.
Mr Mevani also alleged that despite getting such threats in the past on Facebook and Twitter, but the state police took no action.
Rashtriya Dalit Adhikar Manch, a local outfit, announced that Dalit activists will meet the state police chief in Gandhinagar tomorrow to demand 'Y' category security cover for Mr Mevani.