“Mr. Kennedy is the one person who has the qualities that can bring about the unity that most Americans are hungering for,” Mr. Kucinich said. “He speaks a language of conciliation and compassion.”

Besides fund-raising efforts, the Biden campaign has not had much of a public presence since its formal rollout in April. Top officials have spent time in recent days in Wilmington, Del., shopping for office space for a campaign headquarters that is expected to open in July, according to two people familiar with the discussions. The campaign, which has just a few employees on its payroll, is expected to add more staff members once its offices open.