Deshmukh assured the legislative assembly on Thursday that he would make a statement on the issue.
Opposition members demanded that Maharashtra Control of Organised Crime Act be applied against Salman for his alleged links with the underworld.
The Bharatiya Janata Party's Gopinath Munde, who raised the issue in the assembly through an adjournment motion, said the Mumbai police had in their possession tapes of the mobile phone conversation between Salman and Aishwarya in which the actor had threatened her.
In the tape, transcripts of which were published in the Hindustan Times on Thursday, Salman allegedly threatened Aishwarya saying `the underworld keeps tabs on you whenever you are abroad'. Salman is also purportedly quoted as telling Aishwarya that he knew about the Mumbai blasts in advance and that he had links with Dawood Ibrahim.
"Such open threats in broad daylight reflect badly on the law and order situation," BJP legislator Eknath Khadse said.
"The police has taken note of the issue. A statement will be made on the issue after an inquiry," the chief minister said.
"The Aishwarya issue is a welcome change for me going by the tense situation during the last few days," Speaker Babasaheb Kupekar said. It was an apparent reference to the churning in the Shiv Sena legislature group and the crisis in the House over that.