The latest internal government estimates place the cost of adding 40,000 American troops and sharply expanding the Afghan security forces, as favored by Gen. Stanley A. McChrystal, the top American and allied commander in Afghanistan, at $40 billion to $54 billion a year, the officials said.

“The president focused a lot on ensuring that we were asking the difficult questions about getting to an end game here,” the official said. “He knows we cannot sustain this indefinitely.”

Military analysts said the increase reflects a surge in costs for mine-resistant troop carriers and surveillance equipment that would apply to troops in both Iraq and Afghanistan. But some costs are unique to Afghanistan, where it can cost as much as $400 a gallon to deliver fuel to the troops through mountainous terrain.