But lately Mr. Kennedy's message has seemed to shift, including publicly telling a voter who asked about his plans that he was keeping his "options open."

If Mr. Kennedy does decide to leave the party of his famous father and uncles to run in the general election, one potential landing spot may be the Libertarian Party, which at the moment lacks a widely known candidate but has excelled at securing ballot access.

In July, Mr. Kennedy met privately with Angela McArdle, the chair of the Libertarian Party, at a conference they were both attending in Memphis — a meeting that has not previously been reported.