WASHINGTON—President Barack Obama's liberal backers have a long list of grievances. The Guantanamo Bay prison is still open. Health care hasn't been transformed. And Wall Street banks are still paying huge bonuses.

But they are directing their anger less at Mr. Obama than at the man who works down the hall from him. Chief of Staff Rahm Emanuel, they say, is the prime obstacle to the changes they thought Mr. Obama's election would bring.

The...