The nation’s largest veterans groups are urging President-elect Donald J. Trump to keep President Obama’s secretary of veterans affairs, Robert A. McDonald, out of concern that his rumored candidates’ inexperience and ideological leanings could cripple the massive veterans health care system.

On the campaign trail, Mr. Trump was relentless in his criticism of the Department of Veterans Affairs, calling it “the most corrupt agency in the United States.” Some of the candidates mentioned to head the department have called for sweeping changes to make it easier to fire federal workers and privatize veterans’ health care.

But the major veterans groups, which together represent hundreds of thousands of veterans, said they favor the incremental reform they have seen under Mr. McDonald. On Friday, leaders of the American Legion, Disabled American Veterans, Veterans of Foreign Wars, Vietnam Veterans of America and Amvets told members of Mr. Trump’s transition team that they would not support other candidates that have been mentioned.

“We all want McDonald,” said Joe Chenelly, the executive director of Amvets. “He has a good business mind, he is experienced and we feel we can trust him.”

It is unclear whether the Trump team is considering Mr. McDonald, and whether the secretary has met with Mr. Trump. The Trump transition team did not respond to requests for comment. A spokeswoman for Mr. McDonald, Victoria Dillon, said she was not aware if he had spoken to Mr. Trump.

Mr. McDonald, 63, a former Army officer who rose to become the chief executive of Procter & Gamble, was tapped to lead the department in 2014, amid a scandal over long waits for veterans seeking care. Mr. McDonald, who had regularly contributed to Republican candidates, was seen as a business-savvy reformer who could cultivate private-sector efficiency while appealing to both sides of the aisle.

Veterans groups credit him with making smart and steady improvements, while also acknowledging that progress has been slow. Though morale has risen and the health care system has completed millions more appointments each year than it did before Mr. McDonald took over, they say, wait times at many hospitals have yet to improve.

In a meeting with Mr. Trump’s transition team on Friday, veterans groups raised concerns with the names mentioned so far for secretary: Sarah Palin, the onetime vice-presidential candidate; Jeff Miller, the retiring chairman of the House Committee on Veterans Affairs; Scott Brown, the former Massachusetts senator; and Pete Hegseth, who was until recently the chief executive of Concerned Veterans for America, a group funded by the conservative billionaire industrialists Charles G. and David H. Koch.

Veterans groups oppose Ms. Palin and Mr. Miller in part because neither served in the military, Mr. Chenelly said, adding that they think Mr. Brown lacks the experience to run a nationwide health care and benefits system with 350,000 employees.

The groups particularly oppose Pete Hegseth, an Army veteran and Fox News commentator who for years worked for Republican-funded activist groups and think tanks that have portrayed veterans health care as feckless and corrupt.

Currently, about a third of veterans’ health care is provide by private doctors. Mr. Hegseth advocates giving all veterans the choice to seek care in the private sector. Veterans groups say this would siphon billions of dollars away from veterans hospitals, causing the system to collapse.

“If Trump picks Hegseth, it’s going to be war,” said Paul Rieckhoff, the executive director of Iraq and Afghanistan Veterans of America. “He would be a radical departure from what the V.A. has been for generations.”

On the campaign trail Mr. Trump echoed many of Mr. Hegseth’s positions, saying he would give every veteran the choice to see a private doctor. A recent bipartisan study found such a program could cost more than $100 billion over the next decade.

“These guys are like the dog who caught the car,” Mr. Rieckhoff said of Mr. Trump’s transition team. “They suddenly realize how big the machinery really is and that privatization isn’t financially viable. So they are testing names, looking for different ideas.”

Mr. Rieckhoff said the current secretary is a natural fit with Mr. Trump’s vow to bring in outsiders and make business-friendly reforms. “If he wasn’t already the secretary, he would be at the top of his list. He’s a Washington outsider. He’s a C.E.O. of a major company. He’s a Republican. He’s everything Trump would look for.”

Some Republicans and conservative media outlets oppose Mr. McDonald, saying he has been an apologist for the failures of veterans health care, and has not taken needed steps to fire more workers accused of hiding delays and other malfeasance.

Chief among the critics is the small but well-funded Concerned Veterans for America, which Mr. Hegseth left in January. Created by a network of conservative donors organized by the Koch brothers, it sends out regular emails blasting the agency for every failure and misstep.

“Under Secretary McDonald, problems at the V.A. have worsened,” Dan Caldwell, the group’s vice president for policy, said in an email.

He said Mr. McDonald has fired far fewer employees for manipulating wait times than he has indicated in public, and increased employee bonuses, even as the veterans’ health care system has struggled to meet goals.

“The V.A. needs an overhaul,” he said. “Not an extension of the dysfunctional status quo. McDonald’s legacy is one of lies and incompetence.”