Another high-profile Biden administration nominee, that of former civil rights lawyer Julie Su for Labor Secretary, is on the rocks and appears headed for defeat without a vote of the full Senate unless something dramatic happens in the coming days.

In a letter to President Biden, a big bloc of GOP Senators — including the minority leader, Senator McConnell — are urging the administration to withdraw Ms. Su’s nomination before it reaches the Senate floor. The senators accused the nominee of being “consistently evasive” with lawmakers when asked about some of the policies she pursued while head of California’s Labor Department under Governor Newsom.

“In the past three months since she was nominated to be Secretary of Labor, Ms. Su has given senators no indication that her past positions and actions are not indicative of future positions and actions she would take as Secretary,” the letter, signed by 33 Republican senators, states. “She has avoided answering questions whenever possible and she has refrained from providing distinct specificity to her answers when she has responded to inquiries.”

The nomination of Ms. Su, who was most recently a deputy labor secretary, cleared the Senate’s Health, Education, Labor, and Pensions committee on a party line vote in April, but Mr. Schumer, the majority leader, has so far refused to put it to a vote of the full Senate. Senator Manchin has already said he would oppose the nomination, meaning just one more Democratic senator would need to object to scupper her prospects.

Senators Tester and Sinema are the two holdouts who have not said which way they are leaning on the vote. Mr. Tester told reporters on Capitol Hill Wednesday that, “there’s still opportunity to take input so that’s what I’m gonna do.” Mr. Schumer on Wednesday rejected the Republicans’ call for her nomination to be withdrawn. “She is a very good nominee and we are working hard to try to get her confirmed,” he said.

Mr. Biden has held Ms. Su up as a champion of workers’ rights and a key partner of the man she would replace, Marty Walsh. Mr. Walsh left the administration in March to take over as head of the players union for the National Hockey League. “Over several decades, Julie has led the largest state labor department in the nation, cracked down on wage theft, fought to protect trafficked workers, increased the minimum wage, created good-paying, high-quality jobs, and established and enforced workplace safety standards,” Mr. Biden said when he nominated her.

The GOP objections to her nomination revolve around, among other things, her support for a California law that reclassified many independent contractors in that state as full-time employees regardless of whether the employees wanted to make the switch. The law was eventually watered down after considerable criticism about government overreach but remains on the books.

“Laws like California’s arise not from a groundswell of gig worker dissatisfaction but from liberals’ commitment to Big Labor, which would like to see the majority of workers forced to pay union dues,” Senator Thune said in April when her nomination was still in committee. “Ms. Su’s anti-gig-economy, anti-independent-contractor positions aren’t limited to her time in California. During her time at the Department of Labor, Ms. Su has continued to attack independent contracting and gig work.”

In their letter requesting that her nomination be withdrawn, the senators expressed concerns that she would pursue similar policies at the national level. “When asked about her position on this issue and whether she would institute a similar policy as U.S. Secretary of Labor, Ms. Su has been consistently evasive in responding, despite offering strong public support for this policy prior to her current nomination.”