The Scottish Sun said that Derek Mackay contacted the schoolboy over a six-month period, and told him that he was "cute".
Mr Mackay said he had "behaved foolishly" and took full responsibility for his actions.
He also apologised "unreservedly" to the boy and his family.
First Minister Nicola Sturgeon said she did not know about Mr Mackay's "unacceptable" behaviour until Wednesday evening, and is "not aware of any further allegations" against him.
In a statement to the Scottish Parliament, she confirmed that he has been suspended by the SNP while further investigations are carried out.
Police Scotland said it had not received any complaint of criminality, but was "assessing available information from media information". The force also asked for anyone with any information to contact them.
Interim Scottish Conservative leader Jackson Carlaw called on Mr Mackay to also stand down from the Scottish Parliament, saying his behaviour could "constitute the grooming of a young individual".
And Scottish Labour leader Richard Leonard said Mr Mackay's actions had been "an abuse of power" and "nothing short of predatory".
The newspaper detailed allegations that the 42-year-old politician contacted the boy "out of the blue" in August of last year and sent about 270 messages on Instagram and through Facebook.
It has published a list of messages - the most recent of which is from earlier this week - involving Mr Mackay and the boy, in which its says the MSP invited him to dinner and to attend a rugby event.
The newspaper also reported that Mr Mackay contacted the boy several times on Christmas Day, and told him on another occasion that he was "looking good with that new haircut".
In one of the exchanges, Mr Mackay is said to have told the teenager he was "cute". In another the boy confirmed he was 16 and tells Mr Mackay "not to try anything".
The paper also quotes the boy's mother calling for Mr Mackay to be removed from his post.
Chris Musson, the political editor of the Scottish Sun, spoke to the Podlitical podcast about how the newspaper broke the Derek Mackay story, and what the boy and his family wanted to achieve by going public.
In a statement released on Thursday morning, Mr Mackay said: "I take full responsibility for my actions. I have behaved foolishly and I am truly sorry. I apologise unreservedly to the individual involved and his family.
"I spoke last night with the first minister and tendered my resignation with immediate effect.
"Serving in government has been a huge privilege and I am sorry to have let colleagues and supporters down."
Mr Mackay, who has been widely tipped as a future first minister, came out as gay when he left his wife in 2013.
His resignation came just hours before he was due to present the Scottish government's spending plans for the next year - a major set piece event in the Scottish Parliament.
Mr Mackay would have been putting the finishing touches to his preparations to his £43bn budget when he was contacted by the Scottish Sun on Wednesday evening, before the newspaper released the story at 23:20.
The budget was presented instead by public finance minister Kate Forbes.
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