"The Mumbai police will investigate this matter," the home minister added.
Deshmukh said Shiv Sena MLAs Pratap Sarnaik and Sunil Prabhu raised the issue in the Assembly on Tuesday.
"I replied that she had relations with Adhyayan Suman who has said in an interview that she took drugs and also forced him to take drugs. The Mumbai police will look into all this in detail," he said.
Speaking to reporters, Sarnaik denied that this was a political vendetta.
Ranaut had crossed swords with Sena MP Sanjay Raut last week when he said that she should not return to Mumbai if she felt unsafe in the city.
The 33-year-old, Himachal Pradesh born "Queen" star came under all-round fire after she likened Mumbai to Pakistan-occupied Kashmir and made critical remarks against the city police.
"The law is same for everyone. There is no political vendetta. Security is given to people who are looked upon as idols. If there are allegations of drug consumption against her and if they are false, the people leveling them should be thrown into jail," Sarnaik said.
The development comes in the backdrop of actress Rhea Chakraborty's arrest by the Narcotics Control Bureau (NCB) for alleged drug use. The NCB is probing the alleged drugs angle in the death of filmstar Sushant Singh Rajput.
Meanwhile, Brihanmumbai Municipal Corporation (BMC) officials on Tuesday posted a notice at Ranaut's bungalow here, pointing out many alterations undertaken in the building without the civic body's approval, a claim rejected by her.
A BMC team went to the bungalow of the actor in the Pali Hill area of suburban Bandra and posted the notice there as nobody was present to receive it, a civic official told PTI.
The notice pointed out over a dozen alterations in the bungalow, like a "toilet being converted into office cabin" and "new toilets being constructed along side the staircase", the official said.
The BMC has asked Ranaut, who has been provided round-the-clock security by the Centre, to tell it within 24 hours if she had taken approvals for changes in her residence, he said.
Ranaut had on Monday taken to Twitter to share videos of her office premises here and expressed apprehension that the BMC may demolish the property.
The BMC Tuesday filed a 'caveat' in a local court, saying it should be heard first if Ranaut challenges a stop- work notice issued to her.
A caveat is a request to a court that no order should be passed without hearing the person/party which files it.
Ranaut, through her lawyer Rizwan Siddiqui, responded to the BMC's notice, accusing its officials of trespassing on her property and making false claims.
Meanwhile, the actress continued to face criticism for her remarks on Mumbai and its police.
Senior BJP leader Devendra Fadnavis condemned the remarks made by Ranaut and said they amounted to "insult" of the police force.
Ranaut had recently said that she feared Mumbai Police more than the "movie mafia", and would prefer security either from Himachal Pradesh or the Centre.
The Leader of the Opposition said he knew the capacity of the Mumbai police since had had served as chief minister of the state for five years.
"But, the police can come under political pressure," Fadnavis said.
Union minister Ramdas Athawale, an ally of the BJP, said workers of his party RPI (A) will provide protection to Ranaut when she comes back to Mumbai.
"RPI (A) workers have geared up to protect Ranaut who is scheduled to reach Mumbai on Wednesday. Our party workers will offer protection to her at the airport as well as at her residence," Athawale said in a statement.
(Only the headline and picture of this report may have been reworked by the Business Standard staff; the rest of the content is auto-generated from a syndicated feed.)
Business Standard has always strived hard to provide up-to-date information and commentary on developments that are of interest to you and have wider political and economic implications for the country and the world. Your encouragement and constant feedback on how to improve our offering have only made our resolve and commitment to these ideals stronger. Even during these difficult times arising out of Covid-19, we continue to remain committed to keeping you informed and updated with credible news, authoritative views and incisive commentary on topical issues of relevance.
We, however, have a request.
As we battle the economic impact of the pandemic, we need your support even more, so that we can continue to offer you more quality content. Our subscription model has seen an encouraging response from many of you, who have subscribed to our online content. More subscription to our online content can only help us achieve the goals of offering you even better and more relevant content. We believe in free, fair and credible journalism. Your support through more subscriptions can help us practise the journalism to which we are committed.
Support quality journalism and subscribe to Business Standard.