But now the Trump administration appears intent on expanding 401(k) access even more broadly, letting in digital assets such as cryptocurrencies along with private funds, according to lobbyists and other people who work with Wall Street firms on Capitol Hill. They cautioned that the administration’s plans remain in flux and the order could be revised or scrapped entirely.

“The strong signal” from government officials is that the order “will be much more expansive than what was previously forecast by the administration,” said one person who advocates for fund managers in Washington.

The addition of digital assets to the administration’s plans has caused consternation among some private-fund firms, who tried to push back on the idea, say people involved in the discussions. 

“Large private-fund managers would rather not have the crypto distraction,” one said. 

This person added that the private-funds industry still supports the administration’s plans, but is concerned that adding crypto could create a public backlash, and could tarnish private equity’s reputation by linking it with an industry sometimes associated with criminality.

Other private-equity players are less concerned, and are impatient for the order—which has been rumored to be imminent for months—to finally see the light of day, with or without crypto.

The discussions highlight a degree of tension between the crypto and private-equity industries, despite some affinities. Both see the Trump administration as broadly friendly and are hoping to win favorable rule changes in the coming years. 

Crypto has so far been a big beneficiary of the second Trump term. Crypto assets rallied following Congress’s passage of the Genius Act, an industry-supported bill establishing regulatory clarity for digital assets. Trump signed the measure into law July 18. In May, the administration retracted Biden-era guidance discouraging retirement plans from including crypto.

“Allowing regulated, diversified exposure to cryptocurrencies gives individuals more tools to build wealth responsibly, just like they can with other asset classes. As they plan for the future, Americans should be allowed to safely invest in some of the best performing assets of the last decade,” said Summer Mersinger, chief executive of crypto lobbying group the Blockchain Association.

Private equity also has benefited from minor tweaks in regulations over recent months, but is hoping for more significant changes that will let it reach more ordinary investors.

While it is legal to offer private-equity investments to 401(k) plan participants, very few plan sponsors do so, largely out of fear of lawsuits. There are also concerns about how well private-equity funds’ high fees and illiquidity fit with the needs of retirement savers.

Advocates of the change say investing in private assets allows 401(k) participants access to a broader group of companies when far fewer U.S. businesses go public than in the past. They also say it puts defined-contribution savers on an equal footing with defined-benefit participants, whose plans have invested in private equity and other alternative assets for many decades.

Trump’s executive order is expected to direct the Labor Department and the Securities and Exchange Commission to make it easier for defined-contribution plans to offer private funds to participants. The industry hopes for a “safe harbor” provision that would immunize plan sponsors that offer these funds from lawsuits, say lobbyists for the private-equity industry.

But industry critics have opposed the idea.

Sen. Elizabeth Warren (D., Mass.) said July 11 that private funds perform worse and cost more than public funds, and that advocates haven’t explained “why providing retirees with the option to invest their hard-earned life-savings in risky, expensive private markets benefits anyone other than private funds.”

Industry advocates worry that including digital assets in the executive order could intensify this opposition, because of links between crypto and illegal activity, its volatility, and the Trump family’s financial interests in the industry.

Write to Chris Cumming at chris.cumming@wsj.com