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The BBC has asked the Dolly Parton Foundation for comment.
The name would need support from the Metro Nashville Airport Authority board, which is holding a meeting on the issue on 17 September, the governor's office said.
The final decision would require approval from the Federal Aviation Administration.
A Tennessee lawmaker also pledged in recent days to introduce legislation in the statehouse to rename the airport in Parton's honour.
The US has named several airports after famous musicians and other celebrities, including Louis Armstrong New Orleans International Airport and Louisville Muhammad Ali International Airport in Kentucky.
Parton won 10 Grammy awards and sold 100 million records worldwide, including hits like Coat of Many Colors, 9 to 5, Jolene and I Will Always Love You.