Washington (CNN) Veterans Affairs Secretary David Shulkin appears to be on thin ice with the White House, but Shulkin has maintained a critical bloc of support in the nation's veterans groups.

Shulkin will meet with representatives from major veterans organizations on Thursday, two sources confirmed to CNN, as many of their leaders have publicly called on President Donald Trump to keep him at the top of the agency, and worried that a replacement could imperil care for veterans.

Shulkin meets regularly with the heads of the nation's veterans groups.

John Rowan, the head of the Vietnam Veterans of America, defended Shulkin on Tuesday, saying that Shulkin has stayed true to the agency's mission of serving America's veterans -- and that he wants to see Shulkin finish what he started.

"When he came in as secretary, he had a lot of issues no question about it, to deal with, but he's been dealing with it," Rowan told CNN's John Berman. "This is a huge system. If any other system out there was scrutinized like the VA was, they would find problems."

Shulkin ran the Veterans Health Administration under the Obama administration and was unanimously confirmed to lead the VA last year. While his early tenure included a number of bipartisan legislative victories on Capitol Hill, Shulkin has been locked in a power struggle with political appointees within the department, amid a disagreement over to what extent veterans should be able to seek care outside the VA health care system at cost to the government.

Garry Augustine, who heads the Disabled American Veterans' Washington DC headquarters, told CNN he was concerned about the uncertainty surrounding the VA and hoped that Trump will "give the secretary a vote of confidence and we can continue to move forward."

"If you look at the beginning of this administration, some of their biggest victories -- and their only victories -- were VA legislation," Augustine said. "I'd think the President would want to continue in that vein."

The show of support from major veterans service organizations comes as Shulkin's future appears to be tenuous with the White House and the President preparing to oust him. The announcement could happen as early as this week, the source said. However, until an announcement is made, no decision in the Trump White House is considered final.

Several officials told CNN's Jeff Zeleny that Trump wants a replacement for Shulkin to announce seamlessly.

Shulkin has kept a low profile in recent weeks, with few public appearances. Speaking on Capitol Hill earlier this month, he said he was focused on serving veterans, not political distractions.

"I've come here to improve the lives of vets. A lot of people are more interested in politics. I'm interested in getting the job done," he said. "I do believe we are getting back on track. And there is a lot of work to do."

Asked about recent reports that Shulkin's departure was imminent, a VA spokesperson said that there are "no personnel changes to announce at the Department of Veterans Affairs.

"President Trump has made clear that he expects the Department's sole focus to be on providing quality care to America's veterans who have sacrificed to keep this country free and safe," the spokesperson said.

Who could be the next VA Secretary?

At least two potential replacements for Shulkin, confirmed by two sources familiar with White House deliberations, could face tough confirmation fights on Capitol Hill.

One option is "Fox & Friends Weekend" host Pete Hegseth, an Iraq War veteran who previously served as the chief executive of Concerned Veterans for America, a group funded by the conservative billionaires Charles and David Koch. The group played an influential role in the Trump transition and favors, as Hegseth does, giving all veterans the choice to seek care in the private sector.

A White House official on Tuesday was skeptical that Hegseth would be the next leader of the agency.

"I'm not sure about that," the official said of Hegseth, pointing to the skepticism from veterans groups and concerns that Senate confirmation could be a hurdle. The same White House official said there are discussions about replacing Shulkin with someone from the Defense Department.

Hegseth also appears to lack the support of traditional veterans service organizations, who favor a more moderate approach when it comes to downsizing VA and instead relying on a privatized system. Asked if Hegseth would have his support if the President nominated him on Tuesday, Rowan said he would not due to his inexperience.

"No experience in running anything. I mean, the guy's a nice guy apparently, he's a veteran ... but he has no experience running a mega-organization like the VA," Rowan said.

Hegseth has also publicly criticized some lawmakers whose support he would need to win Senate confirmation. In November of last year, he called Georgia Sen. Johnny Isakson, the top Republican on the Senate's veterans panel, a "so-called Republican" and a "swamp creature" amid the Senate's debate to expand the Choice program. More recently, he described the $1.3 trillion spending bill to keep the government's doors open as a "swamp budget."

"This is a Mitch McConnell special. The GOP is not the party of fiscal responsibility," he said on Fox News.

Augustine, who heads up DAV's Washington headquarters, declined to weigh in on specific candidates, but said that "when people come in and start talking about privatization or dismantling the VA ... we have real concerns."

"We believe that the VA is a very good system that needs corrections, needs proper funding and support in order to improve," he said. "It is concerning when we hear names coming up that have called for the VA to be not as stable as it is now."

Another potential Shulkin replacement is former Florida Rep. Jeff Miller, who served eight terms in Congress and was the chairman of the House Veterans' Affairs Committee. He was an early supporter of the President's campaign, and since leaving Congress has worked as a lobbyist, including on veterans issues.

Miller was viewed as a potential pick for VA's top job when Trump first took office, but wasn't in serious consideration because he wasn't a veteran.

CNN and The New York Times previously reported that Trump was considering replacing Shulkin with Energy Secretary Rick Perry, an Air Force veteran and former Texas governor, but Perry told CNN's Rene Marsh that he had no interest in the role and that he is currently in the "perfect job."

While the nation's major veterans groups have nearly unanimously lined up behind Shulkin, others have given Shulkin's tenure mixed reviews.

"Shulkin has received good marks in the first year from veterans' groups and others, but with multiple IG reports and political infighting, it's been tough for him," Paul Rieckhoff, founder and chief executive of Iraq and Afghanistan Veterans of America, told CNN's Wolf Blitzer. "He's had a tough couple weeks. He's under fire right now. Unfortunately, veterans are caught in the middle again."

IAVA has declined to take a position on whether Shulkin should stay on at VA, but Rieckhoff said that "talking to veterans across the country, his stock continues to plummet."