Bronson discussed his proposal at a Tuesday news conference, saying it's too cold for homeless people to spend winter on the streets and he wants to prevent them from dying.
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Bronson also said it's cheaper to send people to warmer climates than pay for housing.
Last year, eight people - a record for the city - died of exposure in Anchorage and the closure of a large arena earlier this year that served as a makeshift city shelter is sure to exacerbate the crisis in a place where winter temperatures regularly dip below zero.
"I have a moral imperative here, and that's to save lives," Bronson said. "And if that means giving them a few hundred dollars for an airline ticket to go where they want to go, I'm going to do that."
If the program moves forward, people can choose to relocate to the Lower 48 or somewhere else in Alaska where it might be warmer or where they have relatives.
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Bronson said a funding source has not been identified, and he's put Alexis Johnson, the city's homeless director, in charge of coming up with a plan for the program.
"Someone says, 'I want to go to Los Angeles or San Diego or Seattle or Kansas,' it's not our business," he said of their intended destination. "My job is to make sure they don't die on Anchorage streets."
The Associated Press contributed to this report.