The annuities being sold, according to people familiar with the matter, are mostly part of a product that proved overly generous to consumers and caused large losses to insurers during the financial crisis. The transaction is complicated and could fall apart without a deal being struck, the people said.

If a deal is reached, it would follow in the footsteps of a pact disclosed by Hartford Financial Services Group Inc. earlier this week involving total consideration of about $2.05 billion. The Hartford deal involves a roughly similar volume of annuity assets. A group of private-equity firms teamed up for the Hartford agreement.

Spokesmen for Voya and Apollo declined to comment.

Both the Hartford deal and the potential Voya one are built around a retirement-savings product known as a variable annuity, which is a tax-advantaged form of investing in stock and bond funds. In the years leading up to the financial crisis, these annuities typically were sold with guarantees of minimum lifetime income if the funds perform poorly.

Voya and Hartford were leading sellers of these type of guarantees in the 2000s, until the financial crisis showed the riskiness of the guarantees to insurers’ own financial health. Hartford had so many guarantees on its books when markets plunged in 2008 that it had to take $3.4 billion in U.S. government aid—since repaid—to help meet regulatory requirements for backing up its obligations to contract holders.

Voya’s then-parent, ING Groep NV, took government assistance in the Netherlands and divested its U.S. life-insurance operations to comply with terms of the aid. Voya took charges against earnings tied to the guarantees. It became an independent publicly traded company over a period of a couple years of share sales by its parent, starting with a 2013 initial public offering.

ING ceased selling variable annuities with the guarantees in early 2010. Since then, Voya and others have been winding down their blocks of business.

Voya’s shares rose 4.5% to $46.15 after The Wall Street Journal reported the talks. Apollo’s shares increased 3.7% to $31.90.

Voya shareholders find the potential deal attractive “because it will let Voya extract some capital from a very low-returning business,” said Colin Devine, a consultant at C. Devine & Associates.

Apollo has experience in the life-insurance industry. After the financial crisis, it teamed with James Belardi, a former senior executive at American International Group Inc., to acquire insurers with annuity businesses.

Mr. Belardi’s focus was fixed annuities, which are a more-conservative type of savings product than variable annuities. With fixed annuities, the insurer promises a specified annual interest rate, or pays interest according to market benchmarks, such as the S&P 500 stock index.

The result of these acquisitions was a company called Athene Holding Ltd. , which went public in December 2016. Apollo earns fees overseeing credit, real estate and other investments that back Athene’s annuity products.

In a Voya transaction, Athene could take on a slice of fixed annuities that would be part of the overall package, according to people familiar with the matter. An Athene spokeswoman declined to comment.

Shares of Athene gained 7.5% to $51.50 on Friday.

Fees from the Athene arrangement have made Apollo’s earnings more stable. Apollo shares have climbed 52% over the past year, outperforming publicly traded peers.

Industry consultants say there might be opportunity for private-equity firms to replicate with variable annuities what Apollo and Athene did with fixed annuities, in rolling up a lot of business into a single firm. Many big U.S. insurers besides Hartford and Voya have older variable annuities on their books with the generous guarantees.

Insurers sell stingier versions these days and charge higher prices for them.

In the Hartford deal, the insurer’s total consideration of about $2.05 billion includes $1.4 billion in cash, a 9.7% stake in the new company, transferred debt and a pre-closing dividend. About 400 Hartford employees will move to the new company, continuing to provide services to the Hartford annuity owners. Similarly, the Apollo negotiations include acquisition of staff to handle the servicing of the Voya contracts, the people said.

Write to Leslie Scism at leslie.scism@wsj.com and Miriam Gottfried at Miriam.Gottfried@wsj.com