Cllr Matheson, who leads the Labour group, had confessed to cheating on his partner and apologised for the “hurt and embarrassment” caused.
The alleged incident took place in the Cathcart area of the city around two weeks before Christmas.
It is understood that on 12 December police were alerted by a passer-by who had stumbled upon two men behaving in an inappropriate manner at Linn View, and found Cllr Matheson performing an ‘indecent act’ with his former partner.
The two men were spoken to by the police who then reported them to the procurator fiscal.
Cllr Matheson is said to have subsequently alerted senior council colleagues that the news would become public.
However, prosecutors decided to take no further action after concluding that there was “insufficient evidence”.
A spokeswoman for Strathclyde Police said: “We can confirm that two men, aged 46 and 38, were the subject of a report to the procurator fiscal in relation to an alleged indecency in the Cathcart area of Glasgow on the 12 of December 2012.”
In a statement, Cllr Matheson said: “I have been having an affair. I have not lived up to my own standards and my loving partner deserves much better.
“I am sorry for the hurt and embarrassment this has caused and I will do all I can to make up for it.”
A Crown Office spokesman said: “The matter was fully considered by independent Crown counsel who concluded that there was insufficient evidence that a crime had been committed.”
Though his party is understood to be standing by the council leader, some have questioned whether Cllr Matheson would be able to cope with the fall-out from incident.
But today a council source said that the council leader was “back at his desk”, and insisted that it was “business as usual”.
The SNP group leader Graeme Hendry declined to comment, saying that the party were focused on the failings of the Labour group, not their individual private lives.
However, Glasgow SNP councillor Jim Torrance took to Twitter to call on people not to judge Cllr Matheson harshly.
He tweeted: “There’s damn few who have never done something, that they are not proud of - ease up a bit on Gordon Matheson.”
The incident has coincided as the council found itself facing growing public protest over its plans to revamp the city’s iconic George Square.
Rumours circulated yesterday that the shortlist of designs for the competition was to be scrapped after it emerged earlier tonight that announcement of the winner, expected to be named today, had been delayed because the jury was unable to come to a decision.
The project has been dogged by criticism from both politicians and the public amid accusations of a lack of consultation and that the process had been rushed. Questions have also been raised about the quality of the designs.
The panel of judges is due to reconvene next week.
One council source said that the ongoing controversy had helped protect Mr Matheson’s position, as nobody wanted to take on responsibility for it.
“I don’t think anyone from within the Labour group will want his job, and that’s why he will survive. He’s there by default. The damage done by George Square is enormous. There’s been a complete breakdown in the relationship with the public; 90 per cent of the calls to the council are against it. They’re going ballistic about this. The council leader’s not got many friends left, and it’s nothing to do with his private life; his judgement’s awful when it comes to the public.”
Cllr Matheson took over from Steven Purcell, who resigned as leader of Glasgow City Council in 2010.
Mr Purcell subsequently admitted he had taken cocaine and feared he would be blackmailed.