The off-the-cuff remark led to a series of meetings within the private equity industry. Ms. Whitmarsh’s approach has been softer than what some have referred to as a “name and shame” strategy. She has sought to make the case that placing more women in decision-making positions will lead to better performance and financial returns.

“I think the business case has become more substantive,” Ms. Whitmarsh said.

Investments are expected to be diversified, so why is that logic not applied to gender, Ms. Whitmarsh asks in the paper she will release at Davos. Men and women have biological and cognitive differences that mean they approach decisions differently, she posits.

Ms. Nyamayaro is trying a different tack. An aim of the HeForShe movement she heads is to get men to champion and voice their concerns for gender equality. When she began her job at U.N. Women several years ago, Ms. Nyamayaro said, she quickly learned that because men hold most positions of power in business and politics, she needed to find a way to get them engaged in the subject.

More important, Ms. Nyamayaro said, she realized that because of all this talk of change, there was no measurement that could serve as a reference point. In the corporate world, for example, companies have historically been reluctant to disclose details of the gender and racial makeup of their employees.

“When we talk about progress, we have to know what it is we’re measuring,” she said. This week, HeForShe will publish for the first time a report of the gender makeup of 10 global companies, including AccorHotels, Barclays, Twitter, McKinsey & Company, Schneider Electric and Unilever.

The 10 companies will disclose the percentage of women in the top 6 percent of executives, the board of directors and new workers. They have also agreed to engage with their employees on the subject, asking them to come up with solutions internally to increase the ratio of female to male employees. In one encouraging sign, more than half of new members to Barclays’ WIN organization — begun as the Women’s Initiative Network but recently broadened to encourage wider participation — were men in 2015, according to HeForShe.

As part of the campaign, HeForShe is also working directly with the chairmen and chairwomen, chief executives or chief operating officers of each participating company, encouraging them to set goals for gender parity. Paul Polman, the chief executive of Unilever, pledged to reach parity in management by 2020. AccorHotels plans to reach pay parity for its 180,000 employees by 2020, Ms. Nyamayaro says.