Dr. Delos Cosgrove, chief executive officer and president of the Cleveland Clinic Health System, speaks during the Cleveland Clinic’s 2010 Medical Innovation Summit in Cleveland, Ohio, on Nov. 1, 2010. (David Maxwell/Bloomberg)

The White House has approached the Cleveland Clinic’s chief executive, Delos “Toby” Cosgrove, a doctor and Vietnam War veteran, about heading the Department of Veterans Affairs, according to an individual familiar with the discussions.

No final decision has been made, according to this individual, who spoke on the condition of anonymity because the White House has yet to formally nominate Cosgrove.

After coming under criticism for weeks over VA’s inability to meet the health needs of veterans — some of whom had to endure long waits for appointments — and alleged efforts by lower-level employees to cover up the problems, VA Secretary Eric K. Shinseki resigned Friday. The agency is being run by his deputy, acting secretary Sloan Gibson.

White House officials declined to comment Tuesday evening.

Eileen Sheil, a spokeswoman for the clinic, would neither confirm nor deny that the administration had asked Cosgrove, nor discuss whether he would accept the job.

The Cleveland Clinic ranks as one of the country’s most renowned medical centers and has won plaudits for the quality of its services and its responsiveness to patients’ needs. The clinic, which Cosgrove has headed since 2004, has a policy of offering same-day appointments to anyone who calls.

Bob Kocher, a former White House adviser on health policy who now does venture capital health-care investments, wrote in an e-mail that if the administration brought on Cosgrove, “they would be recruiting one of the most successful physician leaders and health system operators to come and turn around the VA. Toby only knows how to do health care at a very high level of quality and will not sleep until he instills a similar ethos into the culture at the VA.”

One of the Cleveland Clinic’s doctors, Michael Roizen, wrote an online piece in U.S. News and World Report two weeks ago suggesting Cosgrove was uniquely positioned to help address VA’s current crisis. While he blamed Congress for helping fuel VA’s predicament by failing to budget enough funds, he said smart management could also help right the agency.

“Still, I do know how to solve the problem,” he wrote. “Just get Toby Cosgrove to lead the VA. (And those of us at the Cleveland Clinic hope he doesn’t accept.)”

Obama visited the Cleveland Clinic in 2009, when he was working with Democrats to pass the Affordable Care Act, and he identified it as a model for the nation’s health-care system. Cosgrove, a heart surgeon, also visited the White House a few years ago to discuss the law with the president as part of a small group of health-care industry officials. He also went to the White House last July as part of a session organized by the Business Roundtable.

News of the talks with Cosgrove was first reported by the Wall Street Journal.

Cosgrove served as a surgeon in the Air Force in Da Nang, Vietnam. He was awarded the Bronze Star and the Republic of Vietnam Commendation Medal.

Cosgrove has consulted with top Obama administration officials about the Affordable Care Act, and he has also publicly discussed some of the law’s downsides. In late March he said in a Fox News interview that “about three quarters” of consumers who sign up under the law “find that their premiums are higher than they had been previously with other insurance.”

He also announced last fall the Cleveland Clinic would cut $330 million, a little more than 5 percent of its annual budget, and reduce jobs in anticipation of the law’s impact in 2014.

Lena Sun contributed to this report.