Former Gov. Ben Cayetano has pledged to end the project and replace it with a less-expensive plan that includes dedicated transit lanes and larger buses.
His opponent, former state House Majority Leader Kirk Caldwell, has said the city would be foolish to retreat now.
Critics have blasted the project as a giveaway to engineers and contractors, but supporters say it is integral to reducing travel times in the famously congested island city.
Cayetano beat Caldwell handedly in a four-person primary in August, but the race has narrowed now that they're in the final days of the runoff campaign, according to the newspaper.
The Los Angeles Times reports that the mayoral position in Honolulu is a powerful one. The winner oversees both the city and county. By appointing members of the transit authority and having the power to accept or reject grants, the winner could effectively kill the rail project.