A horse-drawn carriage going through Central Park this week. shannon stapleton/Reuters

New York City Council members continue to waver over a proposed ban on the Central Park carriage horses as Mayor de Blasio’s campaign promise to shut down the industry hangs in the balance.

The majority of the 37 council members who responded to The Wall Street Journal said they were undecided or haven’t yet reviewed the latest version of the legislation, which would phase out the carriage industry by 2016 over the concerns of animal-rights groups, who say the practice is inhumane.

Six council members said they planned to support the bill and seven said they are opposed, while the remaining 24 said they either haven’t made up their mind or haven’t seen the measure in its latest form.

The bill needs 26 “yes” votes to pass in the 51-member council. The bill is being pitched by the de Blasio administration, which sent a draft of the legislation to some council members Monday.

Driver Christina Hansen with her horse, Angelina, along 59th Street near Central Park. John Taggart for The Wall Street Journal

Many council members say they are concerned that banning carriages will cost jobs. The industry employs about 350 people, and the union that represents the drivers, which endorsed Mr. de Blasio last year, has said the jobs aren’t easily replaced.

The new bill backed by Mr. de Blasio includes the creation of a job-training program for the carriage drivers, as well as a provision that would allow the drivers to obtain a license to drive a green taxi, said Councilman Daniel Dromm, who plans to introduce the legislation next week.

Mr. Dromm, a Queens Democrat, said he was confident the bill would ultimately pass.

“I believe that we have the support. I’ve talked with my colleagues in the council and people see this for what it is: an issue of animal cruelty,” Mr. Dromm said.

Jose Abel guides a horse to its stable on 52nd Street in Manhattan on Tuesday. John Taggart for The Wall Street Journal

But others said they viewed the legislation through the prism of job losses.

“Talking about putting 350 families on the street during the holiday season is certainly not very progressive,” said Councilman Costa Constantinides, a Queens Democrat. “It’s a heavily regulated industry. It puts food on the table for 350 families,” he said.

Mr. Constantinides said he doesn’t see the carriages as an animal-rights issue. “We’re in search of a solution without a problem,” he said.

A spokesman for City Council Speaker Melissa Mark-Viverito said she is undecided. A mayoral spokesman didn’t immediately return a request for comment.

Write to Mara Gay at mara.gay@wsj.com