In his address to Congress Tuesday evening, President Trump leaned on some of his standard crowd pleasers: immigration, jobs, terrorism.

But he also revived one of his more surprising proposals, first introduced on the campaign trail last year: “My administration wants to work with members of both parties to make child care accessible and affordable,” he said.

That rhetoric makes Mr. Trump sound more like Hillary Clinton than Ronald Reagan. And a potential debate over child-care policy could offer the rare opportunity for the president and Democrats to cooperate — or at least have a dialogue — over the coming year.

Mr. Trump is not the first Republican president to demonstrate an interest in child-care policy. During his 1968 campaign, Richard Nixon promised to expand access to government-funded day care. But three years later, influenced by the rise of the Christian right, Mr. Nixon vetoed the only universal child-care bill to pass Congress.

Although a few of today’s mainstream Republicans, like Senator Marco Rubio of Florida, have promoted child-care proposals, the official G.O.P. platform does not mention the issue. Conservatives tend to view government-funded child care as an expensive and unwanted intrusion into family life. The position of House Speaker Paul Ryan — whose support Mr. Trump would presumably need to enact a child-care plan — is a case in point. In his 2014 report on poverty, Mr. Ryan fretted over the results of research on Quebec’s public child-care program, which is known for its lax educational standards. Such subsidized care “encourages married women to enter the labor force,” the Ryan report said, leading to “a number of negative behavioral and health outcomes for the children.” (What Mr. Ryan didn’t mention: a competing body of research showing that high-quality day care helps children thrive academically.)

Arguments about the wisdom of working motherhood tend to ignore the fact that working motherhood is the norm. More than half of American mothers work in the year after giving birth, as do 64 percent of women with children under the age of 6, according to the Bureau of Labor Statistics. Working parents require child care, and the typical American family spends 29 percentof its after-tax income on child-care costs, compared with 10 percent or less in many other Western democracies, where child care is provided for or heavily subsidized by the state. Average annual tuition at an American day care center is nearly $10,000, and as much as $30,000 for a high-quality program in cities like New York and Los Angeles.

The issue has been something of a political orphan in recent years. Feminist organizers have focused more on equal pay and paid parental leave, while education reformers have rarely emphasized the period between birth and pre-K enrollment — even though those years are crucial for cognitive development. President Barack Obama’s 2015 plan to increase existing child-care tax credits to a maximum of $3,000 a child from $1,050 went nowhere.

As a presidential candidate, Mrs. Clinton introduced an ambitious proposal to cap child-care costs at 10 percent of a family’s income. But during a campaign season dominated by scandal, Mrs. Clinton’s plan got little attention. Mr. Trump’s tentative forays into child-care policy and paid parental leave — another traditional Democratic issue that he mentioned Tuesday in his speech — have attracted more interest, in part because they are unexpected from a Republican standard-bearer.

In the past, both Mr. Trump and Vice President Mike Pence spoke criticallyabout mothers who work outside the home. But the child-care cause is important to Mr. Trump’s older daughter, Ivanka. In recent weeks, Ms. Trump has met with business leaders and members of Congress to promote the child-care proposal her father first rolled out last September.

Under the plan, individuals earning up to $250,000 a year, and couples earning up to $500,000, would be able to deduct from their taxable income the average cost of child care in their states. The benefit would be modest; for example, a reduction of $840 in federal taxes for a family earning $70,000 a year and paying $7,000 for child care. The plan would offer low-income workers child-care rebates, paid once a year through the earned-income tax credit. The proposal also calls for dedicated savings accounts in which families could invest pretax income to cover child care and elder care costs, as well as incentives for employers to provide child care in the workplace.

Those without wages, like unemployed single parents seeking work or attending job training, would not benefit from the Trump subsidies and are underserved by existing programs. The Child Care and Development Block Grant was created in 1996 as part of welfare reform and was intended to help the poorest parents afford day care. Those benefits currently reach only one out of every 10 eligible children, according to the National Association for the Education of Young Children.

Despite its limited reach, the Trump plan is expensive. An analysis by the Tax Policy Center found it would cost $115 billion over 10 years, most likely making it a nonstarter for Republicans. The proposal also offers little to the working poor, a potential problem for Democratic champions of child care, like Nancy Pelosi and Bernie Sanders. The average annual benefit would be just $10 for families earning $10,000 to $30,000 a year, according to the Tax Policy Center.

It is difficult to square President Trump’s child-care plan with his other budget priorities. Carrie L. Lukas, managing director of the conservative Independent Women’s Forum, supports Mr. Trump’s efforts to help families with child-care costs. But “he also talked about the need for tax simplification,” she said, “which is inconsistent with using deductions” as a social policy strategy.

Still, Ms. Lukas is enthusiastic about some aspects of the Trump proposal, including the fact that married couples with one stay-at-home parent would be able to claim the same tax deduction as many dual-income couples whose children are enrolled in child care. That is an unusual element of the plan; after all, parents who care for their children at home do not incur costs for day care tuition or nanny salaries.

Traditionally, many European nations that enacted government-supported child care had the goal of encouraging maternal employment. More working women means more tax revenue, and better, more accessible child care helps convince parents that they can afford to have more children, who in turn will become future taxpayers supporting the welfare state.

That is not a conservative vision. “It shouldn’t be about pushing to get people into 9-to-5 jobs and kids into day care,” Ms. Lukas said. “That’s not an appropriate role for government. I’ve got five kids, and there are a lot of nonworking parents in my community. They are not only taking care of their own kids, but they are volunteering at school.”

Though she describes herself as a “a libertarian conservative type of person,” Ms. Lukas enrolled her own children in Germany’s government child-care system when the family was stationed in Berlin for her husband’s job. The day care centers “were very expensive for taxpayers,” she said, but she was impressed by the quality of the staff, whose training is subsidized by the German government. “It’s a very serous profession, a very respectable career to pursue,” Ms. Lukas said. “That’s not always the case in America.”

Indeed, the median salary of an American child care worker is about $20,000 a year, a problem the Trump plan does not address.

Elaine Maag, senior research associate at the Tax Policy Center, said the president’s proposal would not “increase the amount of child care available, nor will it increase the quality of care that low-income families will be able to access.” However, Ms. Maag said she was willing to give the president at least a little credit. “I would characterize the plan as identifying an important problem,” she said.