The email from Neera Tanden, who currently helps run Clinton's transition team, is dated March 2, 2015, the day The New York Times revealed that Clinton had used a homebrew email server while serving as secretary of state rather than her secured government account.
While the FBI concluded earlier this year that federal charges against Clinton were not justified in the case, the issue has dogged her campaign to become America's first woman president.
"Why didn't they get this stuff out like 18 months ago? So crazy," Tanden wrote.
Podesta responded, "Unbelievable."
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Tanden later wrote, "I guess I know the answer. They wanted to get away with it."
Tanden suggested the decision to try to keep Clinton's private email use secret could be traced back to Cheryl Mills, a close aide at the time, calling it a "Cheryl special."
"Know you love her, but this stuff is like her Achilles heal (sic). Or kryptonite. She just can't say no to this shit," she wrote.
In the exchange with Tanden, Podesta acknowledged that Mills, Clinton's personal lawyer David Kendall and longtime aide Philippe Reines "sure weren't forthcoming on the facts here."
The Clinton campaign has not confirmed or denied the authenticity of the stream of messages leaked by WikiLeaks, but has accused Russia of directing the hack of Podesta's emails in an effort to tilt the election in favor of Republican Donald Trump.
"Her inability to just do a national interview and communicate genuine feelings of remorse and regret is now, I fear, becoming a character problem (more so than honesty)," Tanden wrote on August 22, 2015.