Republicans in Georgia’s General Assembly have requested a performance review of the top election official in Fulton County, the first step in a possible takeover of the county’s electoral process that could give the Republican-led legislature more control over an area with the largest concentration of Democratic voters in the state.

The request, submitted in a letter on Tuesday by State Senator Butch Miller and signed by about two dozen other Republican state senators, calls for a panel review of Richard Barron, the county election director, over what the lawmakers described as a failure to properly perform risk-limiting audits, a process that helps ensure the correct results and security, after the 2020 election.

“We do so as a measure of last resort, having failed to adequately assuage the concern that we, as elected officials, have regarding the integrity of the Fulton County elections process,” Mr. Miller wrote in the letter.

Fulton County, which includes much of Atlanta, has a record of problems with its elections. Most recently, its June 2020 primary contest was marred by voting machine difficulties that were exacerbated by the small size and poor training of its staff, causing lines to stretch for hours across the county.

But the November general election and the January runoff elections in the county ran relatively smoothly, with few reports of lengthy waits or other complications. There were no legitimate questions about the accuracy of the results in any of the three recent elections. In the presidential race, President Biden carried the county with more than 72 percent of the vote and more than 380,000 votes.

The review process for local election officials is a newly critical element to Georgia elections after state Republicans passed a sweeping new voting law in April. It includes several provisions that lay the groundwork for an extraordinary takeover of election administration by partisan lawmakers.

Under the new law, the State Elections Board is permitted to replace county election board members after a performance review or investigation. But the new law also restructures the state board, stripping the secretary of state of his authority and giving the legislature the ability to appoint members, including the chair.

The letter, which was earlier reported by The Atlanta Journal-Constitution, was signed by three Republican members of the Fulton County delegation in the State Senate. The letter’s authors said they expected members of Fulton County’s House delegation to join them, which would automatically begin the review.

State Representative Chuck Martin, a Republican member of the Fulton County house delegation, said he supported the request for the performance review, and had also spoken to Jan Jones, the speaker pro tempore and another member of the delegation, who supported a review as well.

“Mine is not with an eye on taking over elections,” Mr. Martin said in an interview on Thursday. “This just seems to be the only way we can get data to get answers for the people we represent.”

Mr. Barron, the Fulton County election director, did not respond to requests for comment.

Democrats quickly denounced the move, warning that it undermined the sanctity of future elections.

“After giving themselves unprecedented power under Senate Bill 202, Republicans wasted no time in waging an anti-democratic, partisan power grab, attempting to seize control of elections in Georgia’s largest county, home to the greatest number of voters of color in the state,” said Lauren Groh-Wargo, the chief executive of Fair Fight Action, a Democratic voting rights group based in Georgia. “Their partisan efforts risk election subversion.”

Brad Raffensperger, the Republican secretary of state, supported the review.

“I have called repeatedly for change in Fulton’s elections leadership, so I’m glad Republican legislators are joining me in this effort,” he said in a statement. “After Fulton’s failures last June, I required Fulton to accept a monitor during the general election and runoffs, and forced the county into a consent agreement to start fixing their management problems.”