On social media, many of Mr. Johnson’s backers say they hope that Senator Jeff Flake, the Arizona Republican who has had a public feud with Mr. Trump, might get behind the Libertarian ticket, but so far they have had no luck converting him.

In June, Senator Ben Sasse of Nebraska expressed interest in the Libertarian Party, but later said that Mr. Johnson’s positions on abortion, religious liberty and national security were deal breakers.

Mitt Romney, the 2012 Republican nominee who has showered praise on Mr. Weld, pushed publicly this week for the Libertarians to be allowed in the debates. Still, he has given no sign that he will back the party’s ticket in November, even though he has said he will not vote for Mr. Trump or Mrs. Clinton.

“The Libertarians just break too much from traditional G.O.P. orthodoxy on foreign policy and social issues to get a big amount of Republican support,” said Kyle D. Kondik of the Center for Politics at the University of Virginia.

The Democratic electorate has also failed to provide fertile ground for the Libertarians, despite the lingering disappointment among some who supported Mr. Sanders. Jill Stein, the Green Party nominee, has drawn some Sanders backers, and Mr. Johnson’s position in favor of free trade is out of step with most Democrats these days.

Political analysts point to a variety of reasons third-party candidacies tend to flounder in the United States. A lack of exposure in the news media is often cited, as are political science theories that argue that in a winner-take-all political system, voters gravitate to two mainstream parties out of fear that they would otherwise be wasting their votes.

Many see the debates as the biggest obstacle for third-party candidates. The Commission on Presidential Debates requires that candidates invited to the three nationally televised presidential debates have an average of at least 15 percent support in five specific national polls.