Mr. Trump’s proposal would apply only to new mothers, and he said he would pay for it with savings from reducing unemployment insurance fraud, though it’s unclear how that would work.

The Family Act, a Democrat-sponsored bill stalled in Congress, proposes paying for it with a small increase in payroll taxes. Republicans have previously proposed giving businesses a tax credit for providing paid leave, but making it voluntary.

“If he structures it in a way that it’s not a burden on businesses and does not work against women’s work force participation, I think more Republicans will get behind it,” said Aparna Mathur, a resident scholar in economic policy at the American Enterprise Institute.

Democrats have argued that by limiting parental leave to new mothers, Mr. Trump ignores the role fathers play in raising children, exposes women to discrimination in hiring and neglects the needs of people with ailing or elderly family members.

“It was fantastic to see the Republicans stepping up and having this debate,” said Heather Boushey, the executive director of the Washington Center for Equitable Growth, who was the chief economist for Hillary Clinton’s transition team. “It really spoke to how important these issues are in terms of economic policy. But I hope that the actual policies are really grounded more in the evidence of what works than what he’s put out so far.”

Another policy that analysts say is important to working-class families is paid sick leave, so that workers can care for themselves or for ill family members. It’s unclear whether Mr. Trump will push for it. Thirty-five percent of workers over all and 66 percent of those in the bottom quartile of income get no paid sick leave. Some Republicans have instead proposed letting certain workers — those in the private sector who are not exempt from overtime laws — substitute time off for overtime pay.

Child Care

Another Democratic dream that Mr. Trump raised on the campaign trail was financial help for child care.