McGuire immediately sought out Goodes in the changerooms following the match, apologising for the last quarter incident.
Afterwards, the fuming Magpies president said the fan's behaviour was "despicable".
Follow our live chat for the latest reaction from the football world to the Goodes incident
"I went in to ask Adam what happened, and he told me someone said something; you can ask him what was said," McGuire told AFL.com.au.
"I just apologised to him.
"I wanted to look him in the eye and let him know that we don't stand for this.
"This club doesn't stand for it, the football world doesn't stand for it.
"I didn't want him to go away from being a fantastic leader in what is a wonderful week for our code, or to feel that this is in any way endorsed or condoned, or anything other than that we are just absolutely devastated that someone would say something like that.
"We will be taking every step.
"Everyone knows the rules at Collingwood: if you racially vilify anybody, it's zero tolerance; you're out.
"Hopefully the police have got the person, and we'll find out and get to the bottom of it.
"They should be absolutely ashamed of themselves."
Players took to Twitter to voice their condemnation of the alleged slur and their support for Goodes.
Magpies coach Nathan Buckley echoed his president's sentiments, saying the incident was a reminder that work still needed to be done to rid society of racism, despite gains made in the past two decades.
"My opinion, and it will be the club's opinion, is that any person of any race, colour or creed is welcome to play this game and to observe this game, and should be able to do so with the freedom that any person should be able to," Buckley said.
"That freedom is taken away when you choose not to respect people of other race, colour or creed.
"We're disappointed that this has happened.
"I believe it was a young girl; it speaks probably more to how much more work we need to do in this regard.
"We'll support Adam in this situation, and we'll do anything we need to do to further improve the attitudes in society.
"Football reflects society, and this is a situation that highlights that."
Sydney Swans coach John Longmire welcomed McGuire's visit to the Swans' rooms post-match to apologise, saying that was an indication of how seriously the AFL industry takes racial vilification.
"Adam was clearly disappointed and upset, and he's confident the AFL will work through it," Longmire said.
"He's happy for the AFL to take the lead and let them deal with it."
The incident happened in the Punt Road end forward pocket of the MCG, deep in the final quarter of the Swans' 47-point win.
Goodes reacted angrily when he heard a comment from a female in the front row, pointing her out to security, who removed her from the ground.
After the final siren, the clearly upset dual Brownlow medallist immediately went down the players' race while his teammates celebrated the win on the field.
Neither the Swans nor Magpies have publicly stated what was said.
The incident has come on a weekend of celebration of the contribution of Indigenous footballers, with Indigenous Round marking the 20th anniversary of Nicky Winmar's defiant stance against racism at Collingwood's former home ground, Victoria Park.
The Swans approached the AFL just two weeks ago after Goodes was abused at the same venue in his side's 37-point loss to Hawthorn in round seven. There was no suggestion of racial vilification on that occasion.