NEW YORK—A former president of the United Nations General Assembly and five others were accused Tuesday of engaging in a bribery scheme, part of what federal authorities said was a wider probe into corruption in the top ranks of the international body.

“We will be asking, is bribery business as usual at the U.N.?” said Preet Bharara, the U.S. attorney in Manhattan, at a news conference announcing the charges.

Authorities...