In an effort to end the foreclosure crisis, the Obama administration has been trying to keep defaulting owners in their homes. Now it will take a new approach: paying some of them to leave.

This latest program, which will allow owners to sell for less than they owe and will give them a little cash to speed them on their way, is one of the administration’s most aggressive attempts to grapple with a problem that has defied solutions.

More than five million households are behind on their mortgages and risk foreclosure. The government’s $75 billion mortgage modification plan has helped only a small slice of them. Consumer advocates, economists and even some banking industry representatives say much more needs to be done.

For the administration, there is also the concern that millions of foreclosures could delay or even reverse the economy’s tentative recovery — the last thing it wants in an election year.