The founder of the Abraaj Group, Arif Naqvi, in New York in 2016. He has told investors that he did not misuse their money, pointing to regulatory delays. Ángel Franco/The New York Times

The Abraaj Group has a reputation as one of the developing world’s largest and most influential investors. The private equity firm, based in Dubai, in the United Arab Emirates, manages nearly $14 billion for institutions including development agencies in the United States, Britain and France.

Now some of those investors are claiming that the Abraaj Group and its founder, Arif Naqvi, misused funds, according to people who are familiar with the allegations but not authorized to speak publicly.

The investors — including the World Bank and the Bill and Melinda Gates Foundation — want to know why more than $200 million that they and others had provided was not spent. They fear that Abraaj may have used the money for its own purposes, according to the people briefed on the issue.

Not all of the 24 investors in the fund have complained, people close to Abraaj said.

In discussions with their investors, Abraaj and Mr. Naqvi have said that they did not misuse the money and that there was a misunderstanding about how the fund operates. They said delays in getting approval from regulators to build hospitals in Nigeria and Pakistan prevented Abraaj from deploying funds more rapidly.

The continuing dispute has arisen at a time of flux for the development community. The World Bank president, Jim Yong Kim, and Bill Gates have been promoting the idea that private-sector investors need to take on a larger role in risky projects in the developing world.

In the past, governments and foundations led the way, via direct loans and grants. But over the past year, Mr. Kim and Mr. Gates have argued that it is possible for large pools of capital — such as private equity funds, insurance companies and pension funds — to score big profits by, for example, investing in hospitals in Pakistan and Nigeria. Mr. Kim recently has singled out Abraaj and Mr. Naqvi for praise.

Abraaj, which manages $13.6 billion, is the largest private equity firm dedicated to developing economies. Mr. Naqvi’s fund-raising mantra is: If you want to do good and reap rich, private-equity style returns, invest in Abraaj funds.

More than any of Abraaj’s other offerings, the $1 billion health care fund embodies this ideal. Starting in late 2016, it drew down $545 million from investors to buy hospitals in Nigeria, Pakistan and India. The goal is to improve productivity at the hospitals, allowing them to see more patients — and make more money.

At a meeting in London in September, the investors saw financial statements indicating that over $200 million in cash was sitting with the fund. They didn’t know why the money had not been invested, and they asked the fund’s manager, Khawar Mann, and Mr. Naqvi to see bank statements showing how the money was deployed. They claimed that Abraaj was required to return the money to investors in 60 days if it was not used during that time.

Mr. Naqvi told the angry investors that he saw the company as a vehicle to buy hospitals around the world, and that was why the fund needed the cash on hand. He also cited the regulatory delays. But the investors were not convinced, and he sent more than $100 million back to investors in December.

The investors asked that an auditor with no ties to Abraaj be hired to figure out what had happened. Separately, Abraaj hired KPMG to perform its own audit.

Since September, the investors — which include Proparco, a French development institution, and the CDC Group, a similar body in Britain — have been requesting forensic proof that Abraaj did not use the money to fund its own operations.

Khawar Mann is the manager of Abraaj’s $1 billion health care fund. Tyler Hicks/The New York Times

The Overseas Private Investment Corporation, an arm of the United States government that provides incentives for American firms to invest in developing markets, is exposed to Abraaj via a $150 million loan to the group.

Abraaj and its investors are restricted from publicly discussing matters related to the fund because of nondisclosure agreements signed by all parties.

The World Bank has invested $300 million with Abraaj over the years, including $100 million in the health care fund. The investments were made through the World Bank’s private-sector investing arm, the International Finance Corporation.

The Gates Foundation has invested $100 million in the fund.

When the allegations surfaced, the unit within the World Bank that looks into cases of corruption began an investigation, according to an email exchange between one of its investigators and an outside party that was reviewed by The New York Times. That review closed without finding evidence of wrongdoing, according a person briefed on the review.

Mr. Kim, in his campaign to persuade the private sector to invest more in these types of markets, has often cited Mr. Naqvi and Abraaj as a model.

Just as Abraaj’s fight with investors was heating up in November, Mr. Kim held a video conference with prominent investors from the Middle East at World Bank headquarters. Mr. Naqvi tuned in from Dubai, and as Mr. Kim made his case that the private sector should step up, he repeatedly praised the Abraaj founder.

“Arif has been saying this in so many settings — we are on the same side, screaming this stuff,” Mr. Kim said.

Mr. Naqvi responded in kind.

“Jim, I applaud your leadership,” he said. “We have been partners with the I.F.C. for 10 years, and we are proud that we are one of your larger relationships around the world.”

Even before this controversy, Abraaj’s health care fund was struggling.

The fund has invested about half of the $1 billion it raised from investors. Its largest bet was a $145 million purchase of CARE, a network of private hospitals in Hyderabad, India, and the investment has not met Abraaj’s expectations, according to letters the firm sent to investors last year.

The hospital has been struggling to adapt to an onslaught of regulations from the government. When Abraaj bought the company, it was earning $14 million a year, according to the letters. Now it earns $8 million before taxes and other items. Abraaj has not marked down its value to reflect the lower profits.

The weak performance of the CARE deal has prompted pointed questions on Abraaj’s quarterly conference calls with investors, according to people who have been briefed on the calls.

A Pakistan native, Mr. Naqvi founded Abraaj in 2002, after striking it rich in a private equity deal involving a marketing company in the Middle East. With those profits, he went from being a small regional investor to one of the larger investors in the developing world. He has hired experts in Turkey, in Africa and throughout the Middle East and told them to hunt for deals.

The firm grew rapidly, employing hundreds of people around the world, thanks largely to Mr. Naqvi’s relentless salesmanship. He became a fixture at the World Economic Forum in Davos, Switzerland, and has said he spends 250 days a year on the road pitching pension funds, development institutions and others on his vision of buying companies that benefit society in the world’s poorest countries.

In his most ambitious fund-raising effort yet, he is currently trying to start a $6 billion Abraaj fund focusing on society-improving investments in 30 developing countries.