As desperate chatter about third-party and independent presidential bids grows louder, Gary Johnson, a former governor of New Mexico and a Libertarian presidential candidate, has a message for Republicans who are feeling disillusioned: Give me a chance.

America’s smaller political parties perennially get lost in the shadow of an entrenched two-party system, but as the urgency to stop Donald Trump mounts, interest is growing in some of them. With many Republicans viewing the prospect of Hillary Clinton and Trump on the ballot in November as a dire dilemma, the unorthodox nature of this election year represents a rare opportunity for Libertarians.

“Given the fact that Hillary Clinton and Donald Trump, I think, are two of the most polarizing figures in American politics today, where is the third choice?” Johnson said. “I don’t know how you set the dinner table any more favorably for a Libertarian candidate.”

Johnson, who garnered more than 1 million votes as the Libertarian Party’s nominee in 2012, attracted attention last month after a national Monmouth University poll included him in a hypothetical general election matchup with Clinton and Trump. He drew the support of 11 percent of voters, raising the possibility that if his numbers continue to pick up he could be eligible to participate in the presidential debates this fall.

The Libertarian Party will be on the ballot in all 50 states, and some political strategists have speculated that a mainstream Republican could make a last-ditch effort to stop Trump by trying to run on the Libertarian ticket. Libertarians are unlikely to allow the party to be used as a vessel for such a cause, but they do not mind being part of the conversation.

“I think they’d get their heads handed to them,” Johnson said of any mainstream Republican who sought the Libertarian Party’s backing. But, he added, “it would be terrific from an attention standpoint.”

Patrick Murray, who runs the Monmouth poll, said Johnson remains largely unknown to the general public and attributed his surprisingly strong poll numbers to displeasure with the major-party candidates. He performed best in red states, but overall appeared to siphon slightly more support away from Clinton.

Chagrined about Trump, Republicans, in particular, have been increasingly dropping Johnson’s name when faced with voting for Trump or Clinton.

“I would probably check the box for Gary Johnson, or write in James Madison,” Tim Miller, a former spokesman for Jeb Bush who is spearheading the Stop Trump movement, said last week.

Johnson does not have the Libertarian nomination locked up, but he is widely considered the front-runner ahead of his party’s Memorial Day weekend convention. His opponents include Austin Petersen, the founder of The Libertarian Republic magazine, and John McAfee, who made his fortune developing antivirus computer software and as recently as 2012 was hiding from the Belize police after being sought for questioning in the death of his neighbor.

During a Libertarian forum that aired on the Fox Business Network on Friday night, the three candidates debated issues such as whether a Jewish baker should be required to bake a Nazi wedding cake and pondered questions such as “What is a drug?” and “Why do we have war?” McAfee assured voters that he had never been charged with murdering his neighbor, and Petersen argued that all foreign aid should be stopped.

Although Johnson is turned off by many of Trump’s policies, he can understand the Manhattan businessman’s appeal. When running for governor of New Mexico in the 1990s as a Republican, Johnson used his success as an entrepreneur in the construction industry to make the case that he could bring real-world common sense to state government. He won, serving two terms from 1995 to 2003 and vetoing more than 700 bills.

More recently, Johnson was chief executive of Cannabis Sativa, a company that develops new marijuana products. He quit that job to run for president this year but still uses marijuana recreationally and thinks it should be legalized.

Johnson considers himself a classic libertarian: progressive on social issues and fiscally conservative. He thinks that most Americans are libertarians without knowing it and that in a general election his blend of positions could prove more appealing than people might realize. Hoping to have a better showing than four years ago, Johnson regularly asks voters to take an online quiz that shows his positions have mainstream appeal.

If he does make it to a general election debate, it will be hard for any of his opponents to paint him as “low energy” — as Trump did to Bush. Johnson, 63, is a competitive triathlete who has run 20 marathons, climbed the tallest mountains on all seven continents and almost died twice on gas ballooning adventures. Next year, he plans to challenge himself by riding nearly 3,000 miles on the famed Great Divide mountain bike route, which crisscrosses the Rocky Mountains on dirt roads.

“Unless I am elected president of the United States,” he said.

© 2016 The New York Times Company