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Senior Obama administration officials have agreed that the number of nuclear warheads the U.S. military deploys could be cut by at least a third without harming national security, according to those involved in the deliberations.

Such a reduction would open the door to billions of dollars in military savings, which might ease the federal budget deficit. It also would improve prospects for a new arms deal with Russia before President Barack Obama leaves office, those involved said, but it’s likely to draw fire from conservatives, if previous debate on the issue is any guide.

The results of the internal review haven’t been announced, but they’re reflected in a proposed classified directive prepared for Obama’s signature that details how U.S. nuclear weapons should be targeted against potential foes, according to four people with direct knowledge of the document’s content. The sources, who spoke only on the condition of anonymity because they weren’t authorized to talk to a reporter about the review, described the president as fully on board, but said he hasn’t signed the document.

The document directs the first detailed Pentagon revisions in U.S. targeting since 2009, when the military’s nuclear war planners last took account for the substantial shrinkage – roughly by half from 2000 to 2008 – in the number of nuclear weapons in the American arsenal. It makes clear that an even smaller nuclear force can still meet all defense requirements.

The officials said Obama’s advisers had reached their consensus position last year, after a review that included the State Department, the Defense Department, the National Security Council, the intelligence community, the U.S. Strategic Command, the Joint Chiefs of Staff and the office of Vice President Joe Biden.

Participants said the results weren’t disclosed at the time, partly because of concerns that any resulting controversy might affect Obama’s re-election hopes. Some Republican lawmakers have said they oppose cutting the arsenal out of concern that it might diminish America’s standing in the world.