The incident is understood to have taken place on Friday evening on the town's East Sands beach.
One of those who attended told the BBC that members of the association had "no input" and were "surprised" to discover Obama was on the bonfire.
The St Andrews Conservative Association apologised for any offence caused.Continue reading the main story
It was a stupid thing to do and we apologise for any offence caused”End Quote Matthew Marshall St Andrews Conservative Association The university said it was aware of the reports and was seeking clarification.
Suggestions that the effigy was also wrapped in an EU flag have been denied.
In a statement, the university said it had asked to meet the president of the society to discuss "very understandable concerns".
A spokesman said it would be inappropriate to comment further.
It is understood that the university's discipline officer will be present when the meeting takes place.
Matthew Marshall, president of the St Andrews Conservative Association, said: "President Obama is an important ally to the British government. It was a stupid thing to do and we apologise for any offence caused."'Closest allies'
A former chairman of the university's Labour society described the burning as "disgraceful".
James Mills said: "I can't imagine any other student activists of a major political party would behave in this manner.
"It's disgusting and I hope the Conservative Party and the prime minister completely come out and condemn this obscene act."
He added: "The last thing a truly modern party should be doing is burning an effigy of anyone let alone the first black president of the USA, one of our closest allies."
The same society has burned effigies of Gordon Brown and former South African president Nelson Mandela in the past, along with one of Mr Mills.'Act of hate'
A St Andrews Students Union spokesman said: "As students we believe in political debate, with engaging those who we disagree with us and that all students at St Andrews have a valid opinion to contribute.
"While it does not violate any laws, we believe that the burning of political or public figures in effigy can be an act of hate, stifles productive engagement and can be offensive.
"We believe that St Andrews students should hold ourselves to a higher standard and actively demonstrate that we are ready to engage with, and respect, the opinions of others.
"As such we deplore the burning in effigy of political or public figures regardless of sex, race, political persuasion or any other distinction. "
Sam Fowles, St Andrews Students Union vice president, said: "I do not believe this was a racist act but I don't believe that makes it any less disgusting. Tonight student representatives have shown that the vast majority of St Andrews students are much better than this sort of puerile and offensive behaviour."