The White House’s pick to head the agency that oversees Fannie Mae and Freddie Mac appears unlikely to win Senate confirmation before Congress adjourns due to a sharp policy disagreement between the White House and Senate Republicans over how to regulate the mortgage-finance giants. Senate Republicans are pressing to delay the confirmation of Joseph A. Smith, the North Carolina banking commissioner, to head the Federal Housing Finance Agency. They are concerned he might allow Fannie and Freddie to participate in an Obama administration initiative to write down loan balances, say people familiar with the matter. Mr. Smith first appeared to be headed for a quick confirmation. But he has become tripped up by a broader fight between the White House, which wants to use the firms to help heal housing markets, and GOP critics that say they shouldn’t be run as policy vehicles that create more losses. So far, the firms and their regulator have resisted participating in the program, which is being managed by the Federal Housing Administration and allows banks to write-down loan balances for borrowers who owe more than their properties are worth.