Three of the former staff members were associated with the think tank. Two were directly involved in the dissemination of a press release earlier this week, stating that "as a white male," Rauner would not comment further on the cartoon.
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Rauner has since disavowed the press release.
The controversy followed Rauner as he signed a bill to prevent police from double dipping on pension in Naperville Thursday.
"I have seen it now," Rauner said of the cartoon. "I understand why some people would be upset by it."
Rauner declined to say if he thought the cartoon was racist.
At a second bill signing in Chicago, Representative Lashawn Ford asked Rauner if he would apologize for the cartoon. Rauner declined to do so.
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"I'm not apologizing for anything that I have nothing to do with," Rauner said. "Let me be crystal clear. And that organization that did that cartoon, they should talk to you about what their views are on it, and I'm not gonna."
This is the second time in six weeks that several members of Rauner's staff have resigned or been fired.
"I kinda think some guy that's talked about running on a turnaround agenda, it's becoming quite apparent that it's a turnover agenda," Mayor Rahm Emmanuel said of Rauner's staff changes.
Despite the controversy, Rauner stood with black lawmakers to condemn racism at the Chicago signing.
"With Representative Lashawn Ford, I stand together with him, with our leaders here," Rauner said. "We in Illinois, we in America must stay united against racism, against prejudice, against bigotry against hate speech in all its forms."