The recommendation, in a report by the Education Department and the Consumer Financial Protection Bureau, wouldn’t affect the vast majority of student debt, which is issued by the federal government. It would apply only to the roughly $150 billion, or 15% of total outstanding student debt, issued by private lenders such as SLM Corp.’s Sallie Mae and Wells Fargo & Co.

Consumer bureau chief Richard Cordray said Congress should consider modifying a 2005 law that, except in rare circumstances, prohibits discharging private student loans through bankruptcy.

“It would be prudent to consider whether they wish to modify the code in light of the impact on young borrowers in challenging labor-market conditions,” Mr. Cordray said. He added that the law doesn’t appear to have met its objectives of bringing down borrowing costs and expanding access to private loans.