The Wisconsin Republican leading the state’s partisan inquiry into the 2020 election results on Monday warned election clerks that they would face subpoenas if they did not cooperate and defended the investigation’s legitimacy by declaring that he was not seeking to overturn President Biden’s victory in the state.

“We are not challenging the results of the 2020 election,” Michael Gableman, a conservative former State Supreme Court justice overseeing the investigation, argued in a video posted on YouTube. The inquiry, he said, “may include a vigorous and comprehensive audit if the facts that are discovered justify such a course of action.”

The video from Mr. Gableman comes after he and Wisconsin’s Republican legislative leaders have faced increasing criticism from both their party’s far-right and from Democrats. The right has accused Mr. Gableman of not doing enough to push lies about the 2020 election propagated by former President Donald J. Trump. Democrats have painted the $680,000 inquiry into the election as a waste of state resources and a distraction from other needed business.

Mr. Gableman was assigned to look into Mr. Trump’s false claims that the state’s election was stolen from him by Robin Vos, the Republican speaker of the Wisconsin Assembly, nearly three months ago. The five-minute video released Monday was the first extensive public statement Mr. Gableman has made outlining the scope and aim of his investigation.

The Republicans’ continuing effort to re-examine the 2020 results in Wisconsin comes as Trump allies elsewhere have gone to great lengths to undermine Mr. Biden’s victory. Arizona Republicans are near the end of a monthslong review of ballots in Maricopa County. Pennsylvania Republicans last week approved subpoenas for driver’s license and partial Social Security numbers for every voter in the state. And 18 states, including Texas this month, have passed laws this year adding new voting restrictions.

In recent weeks, Trump-allied conservatives in Wisconsin have shown public frustration at the pace and transparency of Mr. Gableman’s investigation. This month, a group led by David A. Clarke Jr., a former Milwaukee County sheriff who has been a prominent purveyor of false claims about the election, held a rally at the State Capitol in Madison to protest what it argued was insufficient devotion by Mr. Gableman and the state’s Republican leaders to challenging the 2020 results.

Mr. Gableman said on Monday that his investigation would require the municipal officials who operate Wisconsin’s elections to prove that voting was conducted properly. He said local clerks would be required to obey any subpoenas he might issue.

Election clerks in Milwaukee and Green Bay ignored previous subpoenas issued by the Republican chairwoman of the Assembly’s elections committee for ballots and voting machines. Mr. Vos had declined to approve those subpoenas.

“The responsibility to demonstrate that our elections were conducted with fairness, inclusivity and accountability is on the government and on the private, for-profit interests that did work for the government,” Mr. Gableman said. “The burden is not on the people to show in advance of an investigation that public officials and their contractors behaved dishonestly.”

Mr. Gableman added that he did not plan to release information to the public on a regular time frame but would do so when he found it appropriate.

“My job as special counsel is to gather all relevant information and, while I will draw my own conclusions, my goal is to put everything I know and everything I learn before you, the citizen, so that you can make up your own mind,” he said.

The chairman of the Wisconsin Democratic Party, Ben Wikler, said the Gableman video was evidence that state Republicans were at odds with one another over how far the election investigation should go.

“Robin Vos and his far-right ‘investigator’ Michael Gableman are clearly upset that the most extreme fringe doesn’t think they’re going far enough to entertain conspiracy theories,” Mr. Wikler said. “They’re wasting taxpayer funds to serve the political interests of a small group of Republican insiders who want to erode the freedom to vote. It’s a sham, a waste of time and money, and it’s damaging our democracy.”