WASHINGTON — This spring, a longtime staff member for President Obama, Alyssa Mastromonaco, let a friend in on a secret. Mr. Obama had assigned her to begin planning his post-presidential library and foundation.

It was a plum task. Amid a worsening crisis in Syria, early reports of ghosts haunting the HealthCare.gov machine and a dreary sense of second-term setbacks, the foundation glowed with the allure of an eternal Obama afterlife, or at least better days ahead. For Ms. Mastromonaco, the project promised the chance to shape a historic legacy and draft a road map for a 55-year-old former president’s remaining life’s work.

But a few weeks later, White House staff members noticed that an even greater force in the Obama orbit was moving toward the action. Valerie Jarrett, the first couple’s matchmaker, early political patron and undisputed Obama whisperer, sensed a hot property....