But many more borrowers may eventually have their debts set aside. The consumer bureau ordered National Collegiate to hire an independent auditor to review all of its 800,000 loans. The trusts will be prohibited from collecting on any loan on which they cannot prove that the borrower legally owes them the debt.

A recent article in The New York Times revealed that various companies involved in managing National Collegiate’s trusts are missing some of the paperwork needed to legally prove ownership of their loans.

If tens or hundreds of thousands of loans need to be written off, the cost of the settlement could grow far beyond the initial $21 million tally. The insurance company Ambac, which has hundreds of millions of dollars of exposure to National Collegiate’s securities through insurance it sold to investors, warned last month in a regulatory filing that it might need to set aside an additional $200 million to cover losses on its student loan portfolios.

Donald Uderitz, a financier whose private-equity firm is the beneficial owner of National Collegiate’s trusts, said he welcomed the government’s action.

“We’re pleased with the outcome,” Mr. Uderitz, the founder of Vantage Capital Group, in Delray Beach, Fla., said in an interview. “This is independent verification of problems we’ve been investigating ourselves for three years. The audit will allow us to figure out the scope, come up with a compliance plan and make the changes that need to be made.”