A senior police official said the department was committed to provide security individually or collectively to those linked with the film.
"Any individual or group, which attempts to be adventurous, will be dealt with sternly and will face strict legal action," Deven Bharti, Joint Commissioner of Police (Law and Order), said.
The Karni Sena has opposed the Sanjay Leela Bhansali- directed film, alleging it "distorts historical facts" and "hurts sentiments" of the Rajput community.
Karni Sena leader Mahipal Singh Makrana yesterday invoked the nose-chopping incident of 'Surpanakha' in the Ramayana, and said if Padukone does not refrain from fanning sentiments with her provocative language, the Rajputs will not lag behind in acting.
More From This Section
The Mumbai Police yesterday beefed up Padukone's security after the threat.
"Police do not interfere with the protest which is conducted in a democratic manner. We are committed to provide security to everyone individually or collectively and have taken adequate measures for protection of individuals who have received threats," Bharti said.
The Karni Sena has called for a countrywide bandh on December 1, when the film release. In January this year, the Karni Sena had attacked the sets of the movie in Jaipur and even slapped Bhansali.
Recently, through a video statement, Bhansali sought to clear the air surrounding the film, saying he was careful in depicting the "Rajput honour and dignity" in the film.