WASHINGTON—The White House is weighing steps to make it easier for Americans to expunge certain student loans through bankruptcy, opening the door for student debt made by private lenders to be treated on par with credit-card debt and mortgages.

Federal law prohibits student loans, from private lenders and from the U.S. government, from being wiped out in bankruptcy, except in rare circumstances. Other forms of consumer credit such as mortgages, credit-card balances and auto loans face looser requirements for being discharged...