Update: Following the publication of this article, Feinberg made clear that he is staying at Geisinger Health.

David Feinberg, the CEO of Pennsylvania-based health system Geisinger Health System, has told CNBC that he will remain with the firm.

Feinberg was at one point in discussions to lead the Amazon-Berkshire-JPMorgan joint venture to fix health care in the United States, according to people close to the selection process.

However, through a spokesperson, Feinberg now says he remains committed to staying at Geisinger:

"I appreciate being part of the conversation, which I believe reflects the accomplishments of the entire Geisinger team. I personally remain 100% committed to Geisinger and remain excited about the work we are doing and the opportunities ahead as we continue to deliver exceptional care to our patients, our members and our communities."

On Thursday morning, Berkshire Hathaway CEO Warren Buffett and JPMorgan CEO Jamie Dimon announced on CNBC that they had found a final candidate and would publicly name a leader of the venture within two weeks. According to people close to the situation, they did not announce the name on Thursday because Feinberg had not accepted the job.

According to two people familiar with the selection process, the top ten candidates were asked to write a white paper on how they would fix the health care system.

From those white papers, three people were chosen to go through a harrowing interview loop. First, all three talked to Dimon, who referred his two favorite picks to Buffett. Buffett's top choice then interviewed with Amazon CEO Jeff Bezos, who could have vetoed the pick.

Feinberg has been advising the group since the announcement was made in January of this year but ultimately emerged as a top pick to lead the initiative himself, said a person familiar.

Feinberg has been Geisinger's CEO since 2015, after serving as the CEO of the UCLA Hospital System. At Geisinger, he oversees a complex integrated health system that serves more than 500,000 patients in Pennsylvania, which includes 13 hospitals, two research centers and more than 1,600 doctors, as well as various insurance plans. The company claims to be known for its innovation in health care records and health plans, as well as being led by its physicians.

In 2011, Feinberg gave a TEDx talk focused on the importance of improving the patient experience in health care. And he's been an active proponent of increasing access to genetic tests to prevent, and not just treat disease.

The joint venture was first announced in January, and the CEO vetting process has been slow and deliberate.

As CNBC has reported, Todd Combs, an investment manager at Berkshire, has been the lead recruiter for the venture, and venture capitalist John Doerr has also helped out with recruiting.

Some other candidates who have been approached include former Medicare chief Andy Slavitt, former United States CTO Todd Park, and ex-Aetna senior executive Gary Loveman, CNBC previously reported. One of the three finalists was Owen Tripp, CEO of Grand Rounds Health, a start-up that sells a second medical opinion service as a benefit to large employers like Walmart and Target.

Correction: CNBC reported that Feinberg was the top candidate selected to lead the Amazon-Berkshire-JPMorgan joint venture to reform health care. He will not be the CEO of that joint venture.