President Biden’s choice to lead the Labor Department faces an uncertain fate within the Democratic-controlled Senate because of her tenure overseeing a California unemployment agency that paid out billions in fraudulent benefits at the height of the coronavirus pandemic. 

At least three members of the Democratic majority are up in the air on Julie Su, despite having voted to confirm her as undersecretary of labor in 2021. The uncertainty comes as business groups and Republicans have mobilized in opposition to the nominee.

“The buck stops at the top,” said Sen. Mitt Romney, Utah Republican. “The idea of promoting a person who has had that experience to a position of leadership within the entire Department of Labor makes no sense at all.”

The issue with Ms. Su involves her time as California’s labor chief. In that role, she oversaw the state’s disbursement of unemployment benefits as the COVID-19 pandemic shuttered California businesses and forced millions out of work.

With an unprecedented number of people applying for benefits, Ms. Su’s agency was criticized for long wait times and backlogs in processing claims.

According to a nonpartisan report by the California Legislature, more than 5 million Californians had their unemployment benefits delayed during the pandemic. An additional 1 million had their unemployment benefits wrongfully denied.

When her agency got around to disbursing benefits, an independent watchdog found that at least $30 billion was improperly disbursed or stolen by scammers.

“When determining how a person would lead an organization, it’s standard practice to look back at how they have led in the past,” said Sen. Shelley Moore Capito, West Virginia Republican.

Ms. Su has defended her handling of the COVID-era unemployment insurance, arguing that much of the money was approved by Congress without sufficient guardrails in place to prevent fraud.

“The vast majority, over 95%, of the fraud in the unemployment system, not just in California but across the nation, was in the pandemic unemployment assistance program, a program passed by Congress,” Ms. Su said. “The unemployment pandemic assistance program did not have the safeguards that regular unemployment insurance does.”

The defense received pushback from Republicans on Thursday during a hearing on Ms. Su’s nomination within the Senate Labor Committee. Mr. Romney noted that Ms. Su had opted to waive certain oversight and eligibility requirements when administering the pandemic relief program.

Ms. Su said her agency was moving fast to get aid out to as many people as possible and took quick action to deter fraud once made aware of it.

“As soon as we knew there was fraud happening, I shut the door to that fraud. I made changes to the program,” Ms. Su said. “To be very clear, the [California] state auditor never found that my eliminating eligibility requirements resulted in any of the pandemic unemployment system fraud.”

Republicans in opposition to the nomination also note Ms. Su’s championing of a California law that would have required app-based ride-hailing and delivery companies, like Uber and Grubhub, to treat their workers as full-time employees rather than independent contractors.

The law, which was overturned by California voters via referendum, would have forced tech companies to provide paid sick leave and unemployment insurance to their drivers. Republicans say Ms. Su’s support for the state law is problematic, given that the Biden administration is weighing new regulations governing the status of independent contractors.

“If finalized, the new regulation would strip 21 million individuals of their ability to be independent contractors and to enjoy the flexibility this provides,” said Sen. Bill Cassidy, Louisiana Republican. “A law rejected in California is not a policy that we should spread across the nation.”

Senate Labor Committee Chairman Bernard Sanders, Vermont independent, said the attacks on Ms. Su’s record largely come from business lobbyists opposed to any effort to rein in corporate greed.

“She is extremely well qualified,” Mr. Sanders said. “This debate really has everything to do with the fact that Julie Su is a champion of the working class.”

Ms. Su, the daughter of a Chinese immigrant who arrived in the U.S. on a cargo ship, would be the second Asian American to serve as labor secretary.

“When he announced my nomination for U.S. secretary of labor, the president called me the American dream,” Ms. Su said. “My parents believed in it, I benefited from it, and I want to do my part to make sure it is a reality for workers across the nation.”

Democratic Sens. Joe Manchin III of West Virginia, Mark Kelly of Arizona and Jon Tester of Montana have not indicated how they will be voting. Similarly, independent Arizona Sen. Kyrsten Sinema is publicly on the fence.

All four senators backed Ms. Su for undersecretary of labor in 2021. Mr. Manchin has defended that vote, saying he supported Ms. Su largely out of deference to Mr. Biden’s previous labor secretary, Marty Walsh.

Mr. Kelly and Mr. Tester did not return requests for comment on this story. Ms. Sinema does not preview her votes.

With Democrats holding a narrow 51-49 seat majority within the Senate, the opposition could be insurmountable. Adding to the White House’s problems is that Sen. Dianne Feinstein, California Democrat, is suffering from shingles and her return to Washington is uncertain.