Glasgow City Council leader Gordon Matheson, 46, was apprehended by officers after they witnessed him carrying out the act in a car on the city's south side.
The council confirmed police reported the matter to the procurator fiscal but it has been decided no further action will be taken.
Cllr Matheson has been in a long-term relationship with civil partner Stephen Wallace, who was not the other man involved in this incident, which occurred on December 12 in the Cathcart area.
Last night the openly gay council leader, who has recently been co-opted on to Labour's shadow team at the Scottish Parliament, said he had been involved in an affair, adding his "loving partner deserves much better".
Cllr Matheson is understood to have contacted senior figures at Glasgow City Council and within the Labour Party to forewarn them of the news becoming public.
The party is said to be standing by the council leader, who returned Labour to the administration in Glasgow with a significant majority after the local elections in May last year, but sources have questioned whether he can handle the incident.
One said: "I don't see how Gordon will handle this. He's a sensitive guy, a bit thin-skinned and won't like this being in public. Even if Labour stands by him that's different to him having the strength to withstand it."
Another said: "Gordon's been a bit silly here. I'm not so certain it'll impact on his current job but it'll certainly damage his future ambitions. And he's ambitious."
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 spokeswoman for Strathclyde Police said today: "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 12th of December 2012."
A Crown Office spokesman said: "The procurator fiscal at Glasgow received a report in relation to the conduct of a 46-year-old man and a 38-year-old man at Linn View, Glasgow on 12 December 2012.
"The matter was fully considered by independent Crown counsel who concluded that there was insufficient evidence that a crime had been committed."
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