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Mr. Kinzinger, an Air Force pilot, said Sunday, “It is really time for a shake-up,” including Mr. Shinseki’s departure.

Shinseki has been there for six years. He has not been able to get his grip on this,” said Mr. Kinzinger. “The president’s press conference sounded exactly like what he talked about in 2007.”

The congressman praised Mr. Shinseki as a retired Army four-star general but suggested that a hospital administrator might be better equipped to run the Veterans Affairs health care system.

Mr. Shinseki “is a great American, but I don’t think he’s fit for this,” said Mr. Kinzinger. “I haven’t even seen the level of outrage out of him that I think we ought to be hearing. … Just because the general has been in the military doesn’t mean he knows how to run the largest hospital organization in the country.”

Veterans Affairs committee chairmen from both chambers of Congress, while decrying the problems, stopped short of saying Mr. Shinseki had to go.

“You’ve got an entrenched bureaucracy that exists out there that is not held accountable, that is shooting for goals, goals that are not helping the veterans,” Rep. Jeff Miller, Florida Republican and chairman of the House Veterans’ Affairs Committee, told CNN’s “State of the Union.”

Sen. Bernard Sanders, Vermont independent and Mr. Miller’s Senate counterpart, said that while “I think some people may be cooking the books,” that was not something to blame on Mr. Shinseki.

He told CNN that the former general promised to handle wrongdoing and will do so.

Mr. Obama has said he wants to see reports on the issue, including those from the inspector general, but veteran Derek Bennett told “Fox News Sunday,” “There have been plenty of studies.”

“There was a transition team memo from President Bush to President Obama, alleging that there’s some gaming potentially going on,” said Mr. Bennett, chief of staff of the Iraq and Afghanistan Veterans of America. “At some point, we have to stop studying and actually start action.”

The Washington Times reported last week that Mr. Obama’s transition team was warned in 2008 that repeated audits showed the Veterans Affairs Department was misreporting wait times for medical treatment and that the numbers the department gave were not reliable.

The VA has announced that it will pay for some veterans to seek treatment from non-VA hospitals in order to reduce the backlog, which has drawn support from lawmakers.

“We’ve been calling for this for a very long time,” said Mr. Kinzinger. “I introduced a bill similar to this about a year ago, so I’m going to celebrate this move by the Veterans Administration, but I’m also going to say, ‘What took so long?’”