Although no decision has been made yet, Barger has been leading JetBlue since 2007. The company has since become the fifth-largest U.S. carrier.
JetBlue was founded in 1999 and held its initial public offering in 2002. In 2013, the company's stock increased 49 percent, its biggest jump since the IPO.
JetBlue's stock was down by almost 2 percent on Tuesday.
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