Bollywood has also reacted sharply to the comedian's portrayal of the music and cricket icons with many actors slamming the video made by a member of online comedy group AIB, saying it is in poor taste.
Titled "Sachin v/s Lata Civil War", Tanmay, in a video posted on Facebook on May 26, took jibes at the 86-year-old melody queen and the 43-year-old cricketing legend.
DCP (Operations) Sangramsingh Nishandar said a complaint has been given by MNS at Shivaji Park police station with regard to the issue, but the police have not registered any FIR so far.
He said the police have taken a transcript of the video and they would now seek legal opinion from experts before taking further action.
In this regard, police have written to Facebook and YouTube requesting them to block the video and its link, he said.
The DCP said inquiry on the issue is being conducted by the special branch of the city police.The inquiry was initiated after Mumbai BJP president Ashish Shelar complained about the issue to city Police Chief Datta Padsalgikar.
Film wing of Raj Thackeray-led MNS, Chitrapat Sena, has also urged the police to block the video and its link online.
The MNS also threatened to beat him up. Its president Ameya Khopkar, who went to the cyber cell of Mumbai police to lodge the complaint, said "The MNS will not allow Tanmay Bhat to conduct shows in the city, and police should arrest him as quickly as possible."
Taking strong exception to the video, the Shiv Sena asked Chief Minister Devendra Fadnavis to take stringent action against AIB and Bhat for allegedly seeking to vitiate social harmony by mocking the national legends.
Sena leader Neelam Gorhe wrote to Fadnavis seeking action against those behind the video which she said was made by people with "deranged mentality".
"Such people try to misuse the popularity of icons like Sachin and Lata tai for their own publicity," said Gorhe.
"I have also written to Mumbai Police Commissioner seeking action against Tanmay Bhat and AIB," she said.
Shelar said he has spoken to the Police Commissioner and sought action against Bhat and AIB.
Seeing nothing wrong with the comedy video, Bhat in a statement asked those viewers who liked it to email their views to him at a particular ID, stating what they liked.
In a couple of tweets the comedian even reached out to his detractors."Roast wale din yaad aa gaye by god" (It reminds me of our AIB roast days). "Bharat mata ki jai", he said.
Bhat started trending on Twitter with the social media slamming the comedian for his "tasteless" jokes.
Shalini Thackeray, working president of MNS Chitrapat Sena, said she met the officials at the cyber cell.
"I am definitely against the video he posted. It was extremely derogatory. I am at the cyber cell and filing a complaint with them. They are trying to identify under which section he can be booked. I want him to be arrested," Thackeray told PTI.
"He (Bhat) can be booked for defamation, obscenity, targeting women through crude language. I have full faith in the police and the judiciary. Else, we will deal with it in our own way," she said.
She said letting the comedian go without any charges would send a wrong message to the younger generation.
"If you let a person like him (Bhat) get away after insulting such icons, it will send a wrong message to the younger generation. They are on social media today and we need to follow a certain decorum."
"You cannot target icons like Sachin and Lata tai who have brought laurels to the country. What is his claim to fame apart from uttering crude words in the name of humour?"
Thackeray said justifying the video under the garb of "freedom of expression" does not hold true.
"AIB had done this previously as well, with stars like Karan Johar, Ranveer Singh and Arjun Kapoor. We had raised a voice that time too. You cannot pass off anything in the name of humour, democracy, freedom of expression. There should be a limit," she said.
Censor Board member Ashoke Pandit tweeted, "The video by @thetanmay is a classic example of #CulturalTerrorism in the name of humour & FOE. #TanmayRoasted. It's not only about @mangeshkarlata &@sachin_rt but dignity of every common man which should be protected by these so called #Jokers like @thetanmay."