WASHINGTON

Mapping the way out of an unpopular war, the United States and NATO are trying to build an Afghan army that can defend the country after 130,000 international troops pull out. The alliance’s plans for arm’s-length support for Afghanistan will be a central focus of the summit President Barack Obama is hosting Sunday and Monday in Chicago.

The problem with the exit strategy is that someone has to pay for that army in an era of austerity budgets and defense cutbacks.

The problem for the United States is how to avoid getting stuck with the check for $4.1 billion a year.

“This has to be a multilateral funding effort,” said Pentagon spokesman George Little. “We think there should be contributions from other countries.”

That’s partly why so many non-NATO nations are getting invitations to the summit. About 60 countries and organizations are expected to be represented, including nations such as Japan that are far removed from the trans-Atlantic defense pact’s home ground.

More than 20 nations have already agreed to help fund the Afghan army and more are expected to announce their commitments at the Chicago summit. U.S. and other NATO leaders claim that fundraising is on track, although the totals publicly announced so far are small.

A senior Obama administration official said the U.S. and its partners would seek to set targets at the summit for the size and scope of the Afghan security forces after 2014, when foreign forces pull out. The official, who spoke on the condition of anonymity in order to preview the upcoming summit, would not detail pledges expected in Chicago.

That force is now projected to be smaller — and cheaper — than NATO had planned only a year ago. The decision to trim the goal for an Afghan force from about 350,000 to roughly 230,000 was driven more by economic reality than a shift in thinking about Afghanistan’s security needs after 2014, U.S. military officials and NATO diplomats said. The officials spoke on condition of anonymity to discuss internal planning. The larger force had been projected to cost $7 billion a year.

Obama is unlikely to say so, but outside estimates of the U.S. share of the bill for Afghan defense after 2014 range from a quarter to well more than half the total bill. The U.S. will also be on the hook for other support to Afghanistan, but the amount is unclear. The United States is the richest and best-equipped nation in the NATO alliance and long Afghanistan’s largest patron.

Obama signed a pledge with Afghan President Hamid Karzai this month that would obligate the U.S. for a decade.

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