American Airlines passenger service agents are voting on whether to form a union, a process that began two years ago and weathered a bankruptcy filing by American and several attempts by the airline to stop the organizing effort by the Communication Workers of America.

About 9,600 passenger service agents who handle tickets and cargo, work in the VIP lounges, and take customer service calls will start voting Tuesday. They have until Jan. 15 to vote online or by phone.

About 120 agents at Logan International Airport, plus an additional 20 local workers who were laid off in November but still have voting rights, are participating in the election.

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The local workers, along with several community leaders and Boston city councilors, will hold a press conference Tuesday morning at Logan to demand that American obey the law regarding the election. Bruce Hicks, a spokesman for American, said the airline had tried to stop the election from going forward due to a change in the law about the number of signatures required to hold a union vote, but after losing in court, the airline is encouraging all of its eligible employees to vote.

American Airlines is currently restructuring operations through the bankruptcy process. Most of its workers are unionized, including pilots, flight attendants, baggage handlers, and mechanics.