Police Scotland said while it has not “received any complaint of criminality” regarding the SNP politician’s behaviour, it is “assessing available information”.
Meanwhile it emerged that Mr Mackay – who quit the Scottish Government hours before he was due to present the Budget after the scandal broke – has deleted or restricted access to his social media accounts.
His resignation came after The Scottish Sun reported he had sent 270 messages to a 16-year-old he befriended on Facebook and Instagram.
The youngster told the paper: “I was happy to speak to the police and will tell them everything that happened.
“I didn’t think what he was doing was a crime but I knew it was wrong and should be highlighted.”
A Police Scotland spokeswoman said: “We have not received any complaint of criminality, however we are currently assessing available information.
“We would encourage anyone with information to please come forward. Police Scotland will always listen to anyone who wishes to seek advice or formally report a matter to us.”
Mr Mackay – who has also been suspended from the SNP pending investigation – is reported to have called the youngster “cute” as well as offering to take him to a rugby game and out to dinner.
Opposition politicians have condemned what they described as “predatory” behaviour from Mr Mackay – who had been tipped as a future successor to First Minister Nicola Sturgeon – saying the messages amounted to “grooming”.
Mr Mackay, a father-of-two who came out as gay in 2013, has not been seen in public since the story broke on Wednesday night.
It was later reported that the Renfrewshire North and West MSP had also sent dozens of unwanted messages to an SNP activist over a period of four years.
Shaun Cameron, 25, told the Daily Record on Friday that the MSP contacted him on Facebook after meeting him at an SNP event when he was 21. He said some of the messages were “quite suggestive” – alleging the then finance secretary had asked him in September 2017: “Got any naughty pics?”
Ms Sturgeon confirmed to MSPs at Holyrood on Thursday that she had accepted Mr Mackay’s resignation from Government – saying his behaviour had fallen “seriously below the standard required of a minister”.
In his resignation statement, Mr Mackay accepted he had “behaved foolishly” and he apologised unreservedly to the teenage boy and his family.
He said at the time: “I take full responsibility for my actions. I have behaved foolishly and I am truly sorry.”
He remains an MSP but is facing mounting calls to quit.