Still, concerns remain. The new bank, for example, is deciding whether it will give the go-ahead for highly polluting coal-fired power plants, which the World Bank and Asian Development Bank have effectively stopped financing.

Mr. Jin suggested that the bank might make exceptions for poor places where people have no access to power. “Do you leave these people in the dark? It’s a human rights issue,” he said.

When Mr. Xi met President Obama in September, the administration’s icy resolve over the bank had thawed, at least publicly.

Washington has started encouraging the Bretton Woods banks to finance jointly with the Chinese institution. The Chinese, in turn, have pledged to increase their contributions to the World Bank, a sign they will continue to support the existing system.

The Asian Development Bank has already agreed to finance a project or two with the Chinese-led organization. On an October visit to Washington, Mr. Jin was finishing a similar deal with the World Bank.

He is not giving up on the United States, even if the chances are remote.

“We have a standing invitation” for the United States to join the bank, Mr. Jin said, during an appearance at the Brookings Institution. “Anytime you think you are ready, pick up the phone, give me a ring.”

UPDATE: After this article was published, Evan S. Medeiros, a former senior adviser on China for the National Security Council, disputed the account of his conversation with Jin Liqun of the Asian Infrastructure Investment Bank. Mr. Medeiros, who originally declined to be interviewed for the article, said the exchange involving the analogy of buying a cake did not take place. The original source who described the conversation stood by the account.