ATLANTA — Brad Raffensperger, Georgia’s secretary of state, on Monday accused fellow Republicans of trying to undermine the legitimacy of the state’s election in an effort to swing the results to President Trump, who narrowly lost the state to President-elect Joseph R. Biden Jr. and later demanded the hand recount.

Election officials in Georgia also announced Monday evening that they had discovered 2,600 ballots in Floyd County that had not been previously reported to the state, a notable but overall minor hiccup in what they otherwise described as a smooth recounting of the nearly five million ballots cast by Georgia voters during the presidential election.

The counting is expected to wrap up this week, and elections officials have reported few problems aside from the error in Floyd County, which is located in northwestern Georgia and voted heavily for Mr. Trump. Democrats said the recount had so far resulted in no substantive changes, at least none that would affect the lead currently enjoyed by Mr. Biden.

“The Floyd County situation was unfortunate,” said Gabriel Sterling, an official with the office of Georgia’s secretary of state. However, he added, “The majority of the counties right now are finding zero deviations from the original number of ballots.”

Mr. Raffensperger, in an interview with The Washington Post, said that Senator Lindsey Graham of South Carolina was among those Republicans seeking to challenge legally cast absentee ballots with the aim of helping Mr. Trump take the lead in a state in which he is losing by several thousand votes. According to Mr. Raffensperger, Mr. Graham asked about possible ways that ballots could be disqualified, including whether the secretary of state could reject all mail-oinballots in counties that had a high number of signature mismatches on absentee ballots.

A spokesperson for Mr. Raffensperger declined to comment when reached Monday evening. Senator Graham, speaking to reporters on Capitol Hill on Monday, denied that he had suggested to Mr. Raffensperger that he find a way to throw out legally cast ballots.

“That’s ridiculous,” said Mr. Graham, who said that his conversation with Mr. Raffensperger covered questions about Georgia’s system of using signatures to verify the identity of voters who use absentee ballots. “I thought it was a good conversation,” Mr. Graham said. “I’m surprised to hear he characterized it that way.”

Mr. Raffensperger, a lifelong Republican, was stung last week when Georgia’s senators, Kelly Loeffler and David Perdue, both ardent Trump supporters, called for Mr. Raffensperger’s resignation, accusing him of mismanagement and calling the election he oversaw “an embarrassment.”

Mr. Raffensperger has said he would investigate any allegations of irregularities in the election, but insists that the process was sound and the results are valid.

Mr. Raffensperger also hit back at Representative Doug Collins, who is overseeing Mr. Trump’s efforts in Georgia and who accused the secretary of state of caving in to pressure from Democrats. Mr. Raffensperger called Mr. Collins a “liar” and a “charlatan.”

The extraordinary, labor-intensive effort to recount every vote in every one of Georgia’s 159 counties began Friday morning, and counties have until late Wednesday, just before midnight, to complete the work. As of Monday evening, 4.3 million ballots had been recounted, according to the secretary of state’s office, out of just under five million cast.

The secretary of state’s office declined Monday to release the results of the recounts from individual counties. But over the weekend Patrick Moore, a lawyer for the Biden campaign, said that Democrats had been keeping tabs on the county-by-county results and had found that while some discrepancies between the original counts and the recounts had emerged, they were minor and would not affect Mr. Biden’s front-runner status in the state.

“As expected, the counties that have completed their audit thus far have shifted vote totals but almost imperceptibly, and thus far in favor of Joe Biden,” Mr. Moore said during a telephone news conference.

With the newly discovered ballots in Floyd County, Mr. Biden’s lead will go from around 14,200 to around 13,300 votes, according to Mr. Sterling. The New York Times declared Mr. Biden the winner of Georgia’s 16 electoral votes on Friday, joining a number of major news organizations.

Mr. Sterling said that the Floyd County officials discovered the issue in the midst of doing the recount. Mr. Sterling called the error “gross incompetence” on the part of the Floyd County elections director, and said that Mr. Raffensperger had asked the director to step down.

It was the Trump campaign that demanded a hand recount last week in a letter to Mr. Raffensperger. Shortly afterward, Mr. Raffensperger ordered the recount, which his office said is technically an audit.

Even so, Mr. Trump this weekend disparaged the Georgia process on Twitter. “Their recount is a scam, means nothing,” he wrote. On Monday, the president wrote in a tweet that the recount was “meaningless” because “Georgia won’t let us look at the all important signature match.”

This last tweet was a reference to a discredited argument Mr. Trump had made previously concerning a consent decree that helped establish rules for verifying signatures on absentee ballots. But it would be impossible to check signatures in the hand recount because absentee ballots arrive in an envelope with the voter’s signature on the outside of the envelope; those signatures are checked by county elections officials, after which the envelopes are permanently separated from the ballots to ensure voters’ privacy.

The remaining paper ballots, which are being recounted around the state, are thus wholly divorced from the process of signature verification.

Fulton County — Georgia’s most populous county, which covers much of Atlanta — reported that it had completed its recount Sunday; officials noted few problems as they processed roughly 528,000 votes by hand, an effort that involved hundreds of workers who gathered to plow through stacks of ballots in a cavernous hall at Atlanta’s downtown convention center.

Some of Georgia’s other large counties also reported on Monday that the recount was going smoothly. DeKalb County, a populous suburban county on Atlanta’s eastern flank, finished its count of 370,000 ballots Sunday, a process that one county official described as “seamless.”

Ross Cavitt, a spokesman for Cobb County, said that the bulk of his county’s ballots had been counted, but that there was still work to be done Monday. “Nothing that indicates there’s going to be a substantial change in the results,” he added.

Counties that have already certified their vote totals will have to recertify them after the Wednesday deadline if there are discrepancies from the original tally.

The state faces a Nov. 20 deadline for certifying the overall results. After that point, by Georgia law, the second-place finisher may request another recount if the difference in the vote totals is within half a percentage point. (Mr. Biden currently leads by 0.28 percent.)

That recount would be performed by running the ballots through scanners, not by hand.

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