In January 2009, Steve Jobs was sick, gaunt, frail—unable to get out of bed thanks to a painful condition called ascites, a gastroenterological side effect of cancer that caused his belly to swell. The Apple CEO was in desperate need of a liver transplant. He almost found one in the man who would eventually succeed him.

In Becoming Steve Jobs, a new biography by veteran technology reporter Brent Schlender and Fast Company executive editor Rick Tetzeli (due out March 24), it’s revealed that current Apple CEO Tim Cook once offered an ailing Jobs a portion of his liver.

One afternoon, Cook left the house feeling so upset that he had his own blood tested. He found out that he, like Steve, had a rare blood type, and guessed that it might be the same. He started doing research, and learned that it is possible to transfer a portion of a living person’s liver to someone in need of a transplant. About 6,000 living-donor transplants are performed every year in the United States, and the rate of success for both donor and recipient is high. The liver is a regenerative organ. The portion transplanted into the recipient will grow to a functional size, and the portion of the liver that the donor gives up will also grow back.

After going through a series of tests to determine whether a partial transplant was even feasible, he stopped by Jobs’s home in Palo Alto to make the offer; Jobs refused. “He cut me off at the legs, almost before the words were out of my mouth,” said Cook. “‘No,’ he said. ‘I’ll never let you do that. I’ll never do that.'”

“Somebody that’s selfish,” Cook continues, “doesn’t reply like that. I mean, here’s a guy, he’s dying, he’s very close to death because of his liver issue, and here’s someone healthy offering a way out. I said, ‘Steve, I’m perfectly healthy, I’ve been checked out. Here’s the medical report. I can do this and I’m not putting myself at risk, I’ll be fine.’ And he doesn’t think about it. It was not, ‘Are you sure you want to do this?’ It was not, ‘I’ll think about it.’ It was not, ‘Oh, the condition I’m in . . .’ It was, ‘No, I’m not doing that!’ He kind of popped up in bed and said that. And this was during a time when things were just terrible. Steve only yelled at me four or five times during the 13 years I knew him, and this was one of them.”

Stay tuned to Fast Company online and in our forthcoming April issue for exclusive coverage and excerpts from Becoming Steve Jobs you won’t find anywhere else in the days to come.

Interview with Rick Tetzeli, author of Becoming Steve Jobs: