Mr. Buffett won renown and made billions as one of the most successful stock pickers of all time. Mr. Abel’s strengths lie more in running businesses.

Beyond the politics is a brew of resentment and reverence that the president, an Ivy League graduate himself, has long harbored for a club that has never accepted him.

His White House advisers considered giving him such a test to prove his fitness for a second term, but they worried it would draw new attention to his age, according to a forthcoming book.