The union for building maintenance workers, 32BJ SEIU, which has 175,000 members nationally, has also been in contact with the state, a spokeswoman said.
Con Edison has already received reassurance from state officials that its approximately 4,000 to 5,000 customer-facing, critical workers will be prioritized.
“There have been conversations and we understand that we’re on that next group after health care workers,” said Jamie McShane, director of media relations at the energy company.
Tony Utano, the president of Transport Workers Union Local 100, which represents 40,000 city bus and subway workers, said he had talked with Patrick J. Foye, the Metropolitan Transportation Authority chairman who is appointed by Mr. Cuomo, about including his members, and expected that they would be included in the next round.
An advisory panel at the Centers for Disease Control and Prevention has already signaled that it could include a big chunk of the American population: about 87 million essential workers, including police officers, firefighters, teachers, farmworkers, transit workers and others.
The C.D.C. is expected to issue its guidance soon.
In the meantime, companies and groups nationwide have told the federal agency why they should be part of the next wave of vaccinations.
Agricultural workers, from rice farmers to hog farmers, have jockeyed for priority, as have trade associations representing school nurses, truck drivers, morticians and even zookeepers.