Local election officials in Texas have rejected thousands of absentee ballots based on requirements set by the state’s new election law, an alarming jump that risks potentially preventing some Texans from voting in Tuesday’s primary election.

The state’s Republican and Democratic primaries will be the first elections held since the Republican-led Texas Legislature overhauled the state’s election laws. Election officials in the most populous counties have rejected roughly 30 percent of the absentee ballots they have received — more than 15,000 ballots — as of Wednesday, according to a review of election data by The New York Times.

The ballots were rejected largely because voters either did not include their driver’s license number or the last four digits of their Social Security number, or the numbers they put down did not match what officials had on file. The new identification requirements were put in place by the voting law passed last year, known as Senate Bill 1.

The rate of rejection represents a significant increase from past elections, including in 2020, when the statewide rejection rate was less than 1 percent for the general election, according to data from the federal Election Assistance Commission. In 2020, officials rejected 8,304 ballots in Texas out of nearly a million votes statewide. This year, that statewide number has already been surpassed in two counties alone: Harris County and Dallas County rejected more than 8,600 ballots as of Wednesday.

The Times tallied absentee ballot rejections in 10 of the 13 counties with more than 400,000 residents. Bexar County, home to San Antonio, had not started its ballot review process as of Wednesday, and Tarrant County and Denton County, near Dallas, had been delayed by an ice storm.

The total of rejected ballots could still change. Voters have until Election Day to submit their ballots and up to six days to fix ballot defects, depending on the circumstances of the rejections. In Williamson County outside Austin, for example, officials initially rejected 514 absentee ballots, but 167 of those had been corrected and counted as of Tuesday.

The rise in rejections in Texas is the earliest sign that the spate of new election laws passed across the country last year after the 2020 election are having an effect. In the battleground states of Florida and Georgia, Republican legislatures passed sweeping new voting laws with identification requirements for the absentee ballot process that are similar to those in the Texas law. Florida and Georgia will hold their primaries later this year.

Voters throughout Texas have been flooding voter protection hotlines, seeking guidance or expressing dismay that their absentee ballots had been rejected and returned after years of voting absentee without any problems.

At the headquarters of the Dallas County Democratic Party, voters have called in with various issues regarding their ballots. The party has been scrambling to help voters as Tuesday’s Election Day deadline nears, including using text messages to send out information on new requirements to more than 30,000 voters in the county.

“The calls have been pretty much constant since the last week in January, with confusion about the application process and then frustration about the rejections,” said Kristy Noble, the chairwoman of the Dallas County Democratic Party.

Complications over absentee ballots have a more limited impact in Texas than in many other states, however. Texas only allows voters who are over 65 or who have a verified excuse to vote by mail. Though more than a million Texans voted by mail in the 2020 general election, that number is expected to fall this year as turnout regularly dips in the midterms.

But with voting by mail limited to elderly and disabled voters, the concern that initially rejected ballots will disenfranchise voters has grown. Guillermina Nevárez lives at home in the Maverick County border region with her husband, Alfonso Nevárez Sr., and her 98-year-old mother, who is disabled and recovering from a recent surgery.

In all three of their ballots, they missed the field to include their identification information, presuming that since their ballot application had been accepted they were free to cast their ballot.

“We didn’t look at the fine print,” said Ms. Nevárez, who is also the mother of a former Democratic state representative. “And there’s so much of it, the fine print.”

She corrected the three ballots and sent them back by mail. She is hoping that the information is correct — because of her mother’s condition, they cannot go in person to fix any issues.

“It is very upsetting,” Ms. Nevárez said.

The Texas law also bans methods of voting introduced in the 2020 election because of the pandemic, including drive-through voting and 24-hour voting, and it erects new barriers for those looking to help voters who need assistance, such as with translations.

Gov. Greg Abbott, a Republican, signed the law in September. The move came after record turnout in the state: 11.3 million people voted in the 2020 election, including more than nine million who cast their ballots early.

Mr. Abbott’s office did not respond to multiple requests for comment. Previously, the governor’s office has defended the law and blamed the high rejection rate of absentee ballot applications on local election officials.

The Texas secretary of state’s office said that it has been trying to inform voters of the new changes to prevent anyone from having a ballot returned or rejected.

“We have been working around the clock for the past month to get the word out through multiple channels,” said Sam Taylor, a spokesman for the secretary of state, in an email.

The state had already seen abnormally high rates of rejection for absentee ballot applications earlier this month, as voters struggled with the new identification requirements. Now, some voters who had to fix their applications are growing nervous that their ballots will not arrive in time ahead of the Tuesday primary. Others are resolving to just vote in person.

Nancy Bryant, 67, lives in Dallas and has served as an election judge in previous elections. She filled out her application and was approved, so she sent in her ballot. This week, she learned her ballot was rejected and that county officials were going to mail it back to her for corrections.

But with the primary fast approaching, Ms. Bryant had not received her ballot as of Friday, and she’s not sure if she’ll receive it in time to take it to a polling location on Election Day. Without her ballot, she may be forced to cast a provisional ballot. Either way, her wish to cast an absentee ballot has collided with the reality of the new Texas law, and the likelihood of voting in person.

“If I don’t get it back in time, I’m going have to vote provisionally, which hurts me deeply,” Ms. Bryant said. “I am a dedicated voter.”