Americans could stash more in their 401(k)s and sit on their nest eggs longer under a House bill that aims to boost individual retirement savings.

The bill, passed Tuesday by a vote of 414 to 5, raises contribution limits for older workers, and lets companies offer employees a small cash bonus just for signing up for the retirement plan. The bipartisan measure, which some are referring to as Secure Act 2.0, would build on retirement-policy changes enacted in 2019 that, among other things, raised the age people were required to start withdrawing money from retirement accounts to 72 from 70½.

If passed by the Senate and signed into law, the bill would raise the age again over the next decade to 75—a change that would also boost the bottom line of the financial-services industry, which typically earns fees based on the size of retirement accounts. Senators are likely to consider changes to the House bill, and they could then add it to a larger piece of legislation later this year.

For aging people with healthy bodies and healthy bank accounts, the plan would provide significant advantages. In the short term, “It feels like a tax cut,” said Mark Iwry, a senior fellow at the Brookings Institution who oversaw national retirement policy while at the U.S. Treasury Department during the Clinton and Obama administrations.

According to Boston College’s Center for Retirement Research, roughly half of American households are at risk of seeing their standards of living decline after retirement because of a shortfall in savings.

Rep. Kevin Brady (R., Texas), a co-sponsor of the bill with Rep. Richard Neal (D., Mass.), chairman of the House Ways and Means Committee, said it has several aims.

Among them: With “Americans working longer, we want them to keep saving longer,” he said.

What the bill proposes to do

The legislation would gradually increase the age at which savers must start taking withdrawals from 401(k)-type accounts and traditional individual retirement accounts to 73 next year, rising to 74 in 2030 and 75 in 2033. Currently, people who save money in those accounts must begin withdrawing money—and paying any taxes due on it—at 72. These required withdrawals can be a source of frustration for taxpayers who are still working or are trying to make their savings last in retirement.

While the law, if enacted, would help people who can afford to leave their money untouched, it could expose them to higher tax bills in future years. When required distributions kick in, they would be withdrawing more money annually over a smaller time period, said Ed Slott, an IRA specialist.

The increase in the age for required withdrawals “sounds better than it is,” he said.

About 80% of people subject to mandatory retirement account distributions withdraw more than the required minimum because they need the money, said Mr. Slott.

The bill would allow older workers to make bigger contributions. People 50 and older can contribute an extra $6,500 a year to 401(k)-style retirement accounts, for a total of $27,000. The legislation would raise that to $10,000 a year starting in 2024 for people ages 62, 63 and 64. The bill would require catch-up contributions to be made after taxes. Under the legislation, starting in 2024, the extra $1,000 people 50 and older can contribute annually to an IRA would rise to account for inflation.

The bill would generate about $36 billion, according to congressional estimates, to help pay for itself in the next decade, in part by requiring workers ages 50 and older who make extra contributions to 401(k)-style plans starting in 2023 to do so through Roth accounts. These require people to contribute after-tax money, forgoing the tax deductions they would get with a traditional 401(k).

It would also allow employers to give employees the option of channeling matching contributions into a Roth account. Those changes aren’t really tax increases that cover the costs of tax cuts. Instead, the Roth-style accounts just give the government money sooner, during the congressional budgeting window, at the expense of future revenue beyond the next decade.

Sign-up bonuses

Other measures are intended to spur retirement savings among part-time workers, 401(k) participants with student loans and savers with low to moderate incomes.

A measure would make automatic enrollment in retirement savings, which advocates say has been shown to boost participation rates for minorities, mandatory starting in 2024 for newly created 401(k) or 403(b) plans. It would exempt employers with 10 or fewer workers and those in business for less than three years. Those employees would be able to opt out.

It would allow plan sponsors to offer “small immediate financial incentives,” such as cash or gift cards to participants for signing up.

It would also permit employers, starting in 2023, to make matching contributions to the 401(k)-style accounts of employees paying off student loans who don’t contribute enough to the 401(k) plan to receive a full match.

The bill would seek to encourage people with low and moderate incomes to save in retirement accounts by raising the saver’s credit, a match of sorts from Uncle Sam that is often underused. Starting in 2027, the credit would rise to 50% on contributions of up to $2,000 a year, up from a current, tiered structure of 10% to 50% that varies with income.

Another provision would require employers, starting in 2023, to allow part-time employees who work at least 500 hours a year to participate in 401(k) plans after two years on the job, down from three years now.

Retirement lost and found

The bill would require the Labor Department to create an online database through which individuals could search for lost retirement accounts. As Americans change jobs, they are leaving more 401(k)-style accounts and pension benefits with former employers. Some lose track of the money.

Write to Anne Tergesen at anne.tergesen@wsj.com and Richard Rubin at richard.rubin@wsj.com

Corrections & Amplifications

Starting in 2023, a House bill would require employers to allow part-time employees who work at least 500 hours a year to participate in 401(k) plans after two years on the job, a year earlier than permitted today. A previous version of this article contained unclear language, describing the change as “up from three years now.” (Updated on March 30)

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