The decision by Dolan to have the Rockettes perform for Trump sparked a backlash late last month.
After at least once dancer voiced worry about the planned show, the troupe's union sent the dancers a stern email saying "any talk of boycotting this event is invalid" and that "there is no room" for political beliefs in the workplace.
The union, the American Guild of Variety Artists, told the troupe that all full-time dancers, of which there a dozen or so, "are obligated" to dance for Trump.
When that email was publicly disclosed, many people expressed support for the Rockettes not being forced to perform.
Madison Square Garden soon afterward put out a statement saying that no one would be forced to do so, full-time or otherwise.
"It is always their choice," the company said. "In fact, for the coming inauguration, we had more Rockettes request to participate than we have slots available. We eagerly await the inaugural celebrations."
At last week's meeting, on the heels of that comment, Dolan called the inaugural "a great national event," and said, "I think it's very appropriate that the Rockettes dance in the inaugural and the Fourth of July."
"We are an American brand," he said, according to Marie Claire.
The article noted that when one dancer asked if that meant that the Rockettes supported Trump outright, Dolan responded, "No."
"I don't believe it's going to hurt the brand," Dolan also said.
In a statement to CNBC about the Marie Claire story, Madison Square Garden spokesman Barry Watkins said: "This is one person who continues to attempt to represent the entire team of Rockettes." That was a reference to MSG's claim that the article is based on a single source.
"This time it's in a story that is the result of an unauthorized recording that violated the confidentiality of all of her sister Rockettes," Watkins said. "While Mr. Dolan stands behind everything he said during the meeting, no one in that room believed they were speaking publicly. Everyone in the meeting had the chance to speak their mind in a safe setting, and many did."
"Her secret recording was deceitful and cowardly and has betrayed all of her fellow Rockettes," Watkins said. "Posting this story is beneath the ethical standards of Hearst (Marie Claire's publisher). We are deeply disappointed in their decision to post this story."
MarieClaire.com has not responded to CNBC's request for comment on Watkins' statement.
— CNBC's Michelle Castillo contributed to this story.