The race to run Goldman comes at a rare moment of vulnerability for the bank.

Goldman is still highly profitable. So far this year, it has churned out $8 billion in pretax profit. Its stock price values the company at $90 billion. Making partner there remains a yearned-for rank on Wall Street.

But new regulations are clipping profits. Public scrutiny is intense, increasing the likelihood and costs of embarrassing gaffes. Perhaps most important, Goldman’s trading unit, long the engine of its profits, has struggled to adapt to the post-crisis world. In 2009, that business peaked at $33 billion in revenue. This year, the company is on track to make about a third of that.

Mr. Schwartz and Mr. Solomon need to come up with ways to compensate for that slide. In April, they presented their proposals to the board: increase lending and financing activities, expand bond trading and do more banking in cities like Atlanta and Seattle. They calculated that those steps, plus a few others, would generate $5 billion in additional revenue over three years.

The outcome of those efforts is likely to influence who gets picked to lead their firm’s next generation.

From the Gym to the Executive Suite

Mr. Schwartz was born in Morristown, N.J. His father was a scientist. His mother died of cancer when he was 14. Feeling rudderless after high school, he began working as a trainer at a gym. A Rutgers University alumna pressed him to apply to her alma mater — and marched him to campus for an interview after he was turned down initially. He got in on the second try.

“Rutgers was the first place I started to develop my own sense of confidence,” Mr. Schwartz said on a recent walk around the Piscataway campus where he lived for four years, majoring in economics.