THERE’S a place on East 58th Street in Manhattan, with no outdoor sign, where George Soros, David Geffen, Paul Volcker, Jules Kroll and other big names in the business and financial world are regulars.

And it’s not some exclusive private dining room or mahogany-lined boardroom. It’s Sitaras Fitness, which has quietly attracted the upper echelons of Wall Street as members, and in some cases as shareholders.

The center is the brainchild of a bodybuilder, John Sitaras, 39, who along with measuring body fat and biceps has a way of sizing up egos, too. Sometimes, he even juggles the schedules of his 140-plus members to ensure that people who don’t get along in the corporate world are working out at different times.

“He’s sort of a doctor of fitness,” says Jack Welch, the corporate legend who headed General Electric from 1981 to 2001. When Mr. Welch suffered a bacterial staph infection in 2009, he was holed up in the hospital for more than three months and had difficulty walking when he returned home to recover. He turned to Mr. Sitaras to help him get back on his feet.