At the same time, the chairman cautioned that some in the Libertarian Party felt singed by several other candidates joining its ranks in recent years seemingly for the purpose of appropriating its ballot line. They include former Representative Bob Barr of Georgia, a former Republican who was the Libertarian presidential nominee in 2008, and former Gov. Bill Weld of Massachusetts, a Republican who briefly became a Libertarian in 2016 to run as the party’s vice-presidential candidate.

“Those are the two big countervailing forces,” Mr. Sarwark said, referring to Mr. Amash’s political stature and the late hour of his party switch.

The Libertarian nomination is currently set to be decided at a convention next month in Austin, Texas, but the plans for that gathering are now uncertain because of the pandemic. Mr. Sarwark said there would be a conversation among Libertarian Party leaders on Saturday about whether to postpone the convention until the summer, shift to a virtual convention or pursue another option entirely.

Just as fluid is how Mr. Amash would shape the race between Mr. Trump and Mr. Biden. As the Libertarian Party nominee in 2016, former Gov. Gary Johnson of New Mexico won 3.3 percent nationwide, and thousands of critical votes in battleground states. But that was when many right-leaning voters were skeptical Mr. Trump could win and viewed Mr. Johnson as a place-holder for them to not vote for Mr. Trump or Hillary Clinton.

Still, while there may be less of a market for third-party candidates in 2020, Democrats eager to defeat Mr. Trump don’t want to take any chances — especially with a candidate like Mr. Amash who voted to impeach the president.

“I have admired the courage you have shown in standing up to Donald Trump,” Representative Brendan Boyle, a Pennsylvania Democrat, wrote on Twitter on Tuesday night to Mr. Amash. “But your presidential candidacy only increases the possibility that he wins re-election, an outcome that would be catastrophic. For the good of the country, I urge you to reconsider.”