The former finance secretary, 42, stunned the Scottish political world when it was revealed he had sent hundreds of messages to a 16-year-old boy.
The teenager subsequently made a statement to Police Scotland but has not made a formal complaint regarding the behaviour of the politician, who has been suspended by the SNP but currently remains an MSP.
Mr Mackay rocked the Scottish Government when he resigned late on February 5 - just hours before he was due to deliver the 2020 draft budget to MSPs.
Opposition parties have subsequently called for him to resign his seat.
A police spokesman told The Scotsman: “We have not received any complaint of criminality, however, we are currently assessing available information.”
READ MORE: The six most chilling words in Derek Mackay’s messagesIn allegations published earlier this month, Mr Mackay is said to have contacted the boy "out of the blue" and sent about 270 messages on Instagram and through Facebook.
In one of the exchanges it is claimed that Mr Mackay told the teenager he was "cute". In another the boy confirmed he was 16 and tells Mr Mackay "not to try anything".
Mr Mackay, who was widely tipped as a future first minister, came out as gay when he left his wife in 2013. He has not been seen in public since the scandal broke.
A police source told the Daily Record: “It was felt the trail of messages itself was inappropriate given the imbalance in power and the difference in age between the two parties.
“The messages were not highly sexualised and, in any case, the boy was 16 and of age. But consideration was given as to whether the messages could have caused fear and alarm or distress.
“Could there have been an aggravated breach of the peace or would some sort of sexual aggravation or grooming charge be justified? But all things considered, the consensus was that the threshold of criminality had not been reached.”