Many workers can now get time off to volunteer, grieve a death in the family or, more recently, process a miscarriage. Should caring for an ailing pet get an extra paid day off, too?

“ Pawternity leave ”—time off specifically to tend to a new or sick pet—remains a niche workplace perk, but some members of the New York City Council want to change that. Several co-sponsored a bill last month that would let nongovernment workers employed in the city use their paid sick time for pet-related medical care. Given the proposal could apply to millions of employees, it has spurred a broader debate about pet benefits and how many kinds of leave bosses are expected to grant.

“Does that mean we would need to ask the vet for a doctor’s note?” human-resources consultant Teri Chilcoat quipped in a LinkedIn post.

The proposal follows a rush of employee pet benefits in recent years—a reaction to workers’ demands for more work-life balance, not to mention the millions of Americans who adopted dogs, cats and other animals during the pandemic. Companies such as United Airlines and Verizon provide discounted pet insurance while some, including Adobe and Lyft , offer access to subsidized online vet calls via pet telehealth services like Airvet.

The job market has since cooled, though, and employers say there’s a limit to how far they’ll go to accommodate their workforces’ out-of-office needs. In the new climate, some bosses say granting paid sick time for employees’ furrier family members is one benefit too far.

‘Take care of our people first’

“Needing an hour or two to run to the vet is one thing, but it is not fair to the other employees on the team for someone to be out because of a personal decision that their co-worker made,” says Ross Snyder, president of White Tie Productions, an event-production company in Phoenix with 15 full-time employees.

His company has an unlimited paid-time-off policy that includes sick time. Yet using it to take care of a pet would have to be approved by a manager, who may decide not to grant it, he says.

There are signs companies are beginning to dial back pet-related benefits. After rising in recent years, the number of employers offering pet insurance fell to 13% in 2024 from 16% in 2023, according to a MetLife survey of more than 2,500 companies.

“Companies are looking to cut costs so maybe they drop health benefits that are more peripheral like pet insurance,” says Andy Challenger, senior vice president of outplacement firm Challenger, Gray & Christmas. In a Challenger survey of 182 companies this summer, more reported they were trimming benefits to cut overall costs than said so in 2023.

“As a pet owner, I support [paid pet leave], but as an advocate for employers, we need to take care of our people first,” says Jon Hyman, a partner at Wickens Herzer Panza in Avon, Ohio, who advises privately owned businesses on employee matters. Many workers still don’t have paid sick leave, he says, and any legal mandate on pet care benefits is bound to drive up companies’ administrative costs.

“It’s the legislative hammer that is giving me ‘pause’ here, no pun intended,” he says.

Time off for potty-training

Nearly two-thirds of Americans own a pet, and half of pet owners say their pets are as much a part of their family as a human member, a 2023 Pew Research Center survey shows. In contrast, the number of households with children under 18 has fallen to 39% from 48% in 2000, according to 2023 U.S. Census Bureau data.

Annika Cha joined New York-based legal search marketing firm 9Sail last year. The company offers two days’ leave, which it will soon raise to three, to help get new pets settled in at home. That helped persuade her to adopt a dog, she says.

“I was like, that is so cool,” she says. Having the days off, she adds, was key to quickly potty-training her new mini poodle, Ollie.

Councilman Shaun Abreu, who introduced the New York City proposal, says one of its aims is to promote pet ownership and boost New Yorkers’ mental health.

The bill wouldn’t mandate additional paid time off for pet-owning workers but extend the city’s existing sick-time laws. Companies with more than 100 employees are currently required to provide those working in the city up to 56 hours paid annual leave to take care of themselves or a sick family member. Businesses with fewer than a 100 staff must provide up to 40 hours.

Some business owners say they’ve discovered firsthand the benefits of pet-related time off.

When her beloved orange tabby, Maud, died in October, Samantha Brooks took two days off.

The 38-year-old, who owns Taro Financial, a small financial-services firm in Colorado Springs, Colo., spent the mini break hiking nearby mountains and taking “some very quiet time.”

Upon her return, she decided to institute a pet-bereavement leave policy for her staff. They now get up to three paid days off to grieve the loss of a pet, in addition to their regular paid time off.

“Anyone who has lost a pet will tell you that it was one of the worst days of their life,” she says.

Write to Ray A. Smith at Ray.Smith@wsj.com