Later that spring, on May 3, 2011, Mr. Gates again visited Mr. Epstein at his New York mansion, according to emails about the meeting and a photograph reviewed by The Times.

The photo, taken in Mr. Epstein’s marble-clad entrance hall, shows a beaming Mr. Epstein — in blue-and-gold slippers and a fleece decorated with an American flag — flanked by luminaries. On his right: James E. Staley, at the time a senior JPMorgan executive, and former Treasury Secretary Lawrence Summers. On his left: Mr. Nikolic and Mr. Gates, smiling and wearing gray slacks and a navy sweater.

A Vast Charitable Fund

Around that time, the Gates Foundation and JPMorgan were teaming up to create the Global Health Investment Fund. Its goal was to provide “individual and institutional investors the opportunity to finance late-stage global health technologies that have the potential to save millions of lives in low-income countries.”

As the details of the fund were being hammered out, Mr. Staley told his JPMorgan colleagues that Mr. Epstein wanted to be brought into the discussions, according to two people familiar with the talks. Mr. Epstein was an important JPMorgan customer, holding millions of dollars in accounts at the bank and referring a procession of wealthy individuals to become clients of the company.

Mr. Epstein pitched an idea for a separate charitable fund to JPMorgan officials, including Mr. Staley, and to Mr. Gates’s adviser Mr. Nikolic. He envisioned a vast fund, seeded with the Gates Foundation’s money, that would focus on health projects around the world, according to five people involved in or briefed on the talks, including current and former Gates Foundation and JPMorgan employees. In addition to the Gates money, Mr. Epstein planned to round up donations from his wealthy friends and, hopefully, from JPMorgan’s richest clients.

Mr. Epstein thought he could personally benefit. He circulated a four-page proposal that included a suggestion that he be paid 0.3 percent of whatever money he raised, according to one person who saw the proposal. If Mr. Epstein had raised $10 billion, for example, that would have amounted to $30 million in fees.

Ms. Arnold said Mr. Gates and the foundation had been unaware that Mr. Epstein had been seeking any fee. She said Mr. Epstein “did propose to Bill Gates and then foundation officials ideas that he promised would unleash hundreds of billions for global health-related work.”