A group including former elected officials, former cabinet members and others is pressuring the commission that sets the rules for presidential debates to make it easier for third-party candidates to join, the latest challenge to the requirements for participation.

The Commission on Presidential Debates, a nonprofit corporation set up by both major parties in the late 1980s to organize the debate process, restricts participation in debates to candidates who have won at least 15% support in five national polls. The rule, instituted...