To avoid consolidating more than $5 trillion in assets and liabilities onto the federal ledger, the government never took full control of Fannie and Freddie when it bailed them out. Instead, the U.S. agreed to inject vast sums of aid in exchange for a new class of stock-'senior preferred' shares-that paid a 10% dividend. The government also received warrants to acquire nearly 80% of the firms' common stock.

The shares, considered worthless, were delisted from the New York Stock Exchange in 2010. But soon after, some investors concluded the firms would become profitable one day and began buying their shares at deep discounts. The government upended those bets last year when it amended the terms of its rescue.

The new agreement requires all of the firms' earnings to be sent to the Treasury as dividends. Those payments don't reduce the senior preferred shares held by the Treasury, however, and they prevent the firms from building capital. As a result, restoring them to private ownership requires Congress or the Treasury to change the bailout terms.

Hedge funds have taken to Capitol Hill to build support. In a June letter to Treasury Secretary Jacob Lew, Rep. Michael Capuano (D., Mass.) called the current bailout terms 'outrageous usury' and introduced a bill that would relax them, making it more likely that Fannie and Freddie could one day exit government control.

But shareholders have found mostly an unreceptive audience. Taxpayers should be entitled to every penny of profit for so long as Fannie and Freddie are viable due solely to their government support, said Mr. Corker, the Tennessee Republican. His bill would leave little for shareholders.

'These entities are totally worthless from the standpoint of generating any kind of [income] without the federal government's' backing, he said in July. 'For the sake of the taxpayers, I hope that you do not win. But you have at it.'