By Anne Tergesen

Sept. 23, 2015 7:00 a.m. ET

A giant in the business of guiding 401(k) plan savers is making a push to offer millions of those investors access to its human advisers as well as online tools.

Financial Engines Inc. hopes the move will appeal to employers looking to help workers achieve “financial fitness,” while also enticing more individual savers to hire the Sunnyvale, Calif., firm to manage their 401(k) money for a fee.

The development comes at a time when many financial companies are looking for ways to combine computerized and human assistance, as automated “robo advisers” challenge traditional one-on-one advice relationships. Vanguard Group recently launched a service that includes both Internet-based advice and contact with a salaried Vanguard adviser.

Roughly nine million people currently have access to Financial Engines’ services through an online tool, which is paid for by fees charged to plan sponsors.

Starting next year, about half of those people will be invited to speak to advisers via telephone, webcam or live chat for free, the company said Wednesday.

Advisers can help callers validate their investment strategies, figure out how to balance a budget, decide when to claim Social Security, and understand the impact on a nest egg of raising or lowering their savings rate, Financial Engines spokesman Mike Jurs said.

Kelly O’Donnell, an executive vice president at Financial Engines, said the new service is designed, in part, to appeal to plan sponsors seeking to add so-called financial-wellness assistance. Modeled after the physical-wellness programs that aim to help employees lose weight or improve their physical fitness, financial-wellness offerings can feature a range of programs, including counseling sessions designed to help workers better manage their finances, such as by balancing a budget or paying down debt. A number of employers—such as Meredith Corp., Staples Inc. and PepsiCo Inc. --are introducing such services.

“Our clients assess us on how much impact we are having,” said Ms. O’Donnell. She said the questions they ask include, “Are participants saving more? Are their portfolios better diversified? Will they have enough for retirement?”

To improve such outcomes, “we want to be able to help all 401(k) participants look at their full financial pictures,” Ms. O’Donnell said.

Along the way, she said, Financial Engines hopes that more 401(k) participants will decide to enroll in—and pay for—its managed-account service, which includes deciding how much to invest in stocks and bonds, choosing from among the plan’s investment options, and periodically rebalancing an investor’s account to its target asset mix.

Financial Engines is already the leader in the fast-growing market for 401(k) managed accounts. According to Cerulli Associates, a research firm that specializes in the asset-management industry, Financial Engines had $114.5 billion in assets under management at midyear, or 60% of the $186.7 billion market. Rivals Morningstar Inc., Fidelity Investments and GuidedChoice Inc. had respective market shares of 20%, 7% and 7%.

Having a managed account gives investors professional help with their retirement investments, often at a lower cost than they would pay for a traditional financial adviser. Customers generally work with a call-center team rather than a particular adviser, and managed-account staff don’t typically provide help with broader financial issues like insurance or estate planning. Financial Engines says clients can request a specific adviser.

According to Cerulli, the $186.7 billion in managed accounts is up from $124 billion in 2013. Benefits consultant Aon Hewitt said about 52% of large 401(k)-plan sponsors offered managed accounts in 2013, up from 29% in 2011. Many industry watchers expect demand for the services to grow as the baby boomers get closer to retirement—frequently a time when employees seek out the more customized advice available through a managed account.

Managed accounts compete with target-date funds, all-in-one mutual funds that automatically reset an investor’s asset allocation by reducing stock holdings and increasing the portion in bonds as the investor ages. But while many more 401(k) participants use target-date funds than managed accounts, some criticize the funds for taking a one-size-fits-all approach to the asset allocations of investors who are the same age.

Financial Engines’ fees range from 0.2% to 0.6% of assets a year for the managed account service, in addition to the fees of the mutual funds 401(k) participants invest in. The exact fee depends on the size of an individual’s account as well as the size of the plan they belong to.

Currently, the managed-account program is available to nine million employees in 650 plans, and about 10% of the participants are paying for the service.

Financial Engines and other advisers are also offering services to help retirees draw down a steady income after they have left their jobs. Some, including Financial Engines and GuidedChoice, provide income-planning and investment-management services that were previously available only in 401(k) plans. The companies may also help workers figure out the optimal time to claim Social Security benefits.

Write to Anne Tergesen at anne.tergesen@wsj.com