Mr. Prince appears increasingly likely to challenge John Barrasso, a senior member of the Senate Republican leadership, according to people who have spoken to him in recent days. He has been urged to run next year by Stephen K. Bannon, who is leading the effort to shake up the Republican leadership with financial backing from the New York hedge fund billionaire Robert Mercer and his daughter Rebekah.

Over the weekend, Mr. Prince traveled to Wyoming with his family to explore ways to establish residency there, said one person who had spoken to him.

If he runs, Mr. Prince would face formidable obstacles in seeking to unseat Mr. Barrasso, a popular and genial but low-profile senator who will have the full backing of Senator Mitch McConnell, the majority leader, and the well-funded political committees loyal to him. Mr. Prince, who has never run for public office, has been a polarizing figure for years, as Blackwater faced a welter of ethical and legal problems over its work for the military in places like Iraq, including an episode in 2007 in which its employees killed 17 civilians in Baghdad.