At the heart of the debate is the future of America’s relationship with Russia. With Mr. Trump fighting accusations that his associates might have colluded with Russian officials during the election, administration officials acknowledge that it is almost impossible to imagine a new round of arms control negotiations that might ease the need for a major buildup. The Russians are still building, and the United States has accused Moscow of violating an intermediate-range missile treaty, forcing Washington to develop a response.

But also on the table are other revived nuclear weapons, all under the control of the Energy Department, as well as the really big-ticket military items: a stealthy nuclear bomber to replace the B-52 and B-1 bombers, and a fleet of new, silent submarines. Most controversial are plans to overhaul the oldest and most vulnerable part of the American nuclear complex: the Minuteman missiles that are buried in silos across the Midwest and West. The Pentagon conceded last year that the missiles are so antiquated that they are still run on eight-inch floppy computer disks.

Upgrading the missiles would be among the most expensive parts of a Trump military buildup, and critics say it is time to give them up.

“There are ways to save money for the country that do not in any way put us at risk,” said Tom Z. Collina, director of policy for Ploughshares Fund, an independent organization that favors nuclear arms control, “because many elements of this program are excessive, redundant and dangerous.”

Mr. Collina called the budget office estimate “very credible” and consistent with earlier calculations of the program’s cost.

As Mr. Obama’s initiative becomes Mr. Trump’s, “there is going to be a backlash coming,” particularly among Democrats who had supported the program, said Jon B. Wolfsthal, who oversaw nuclear issues in the Obama White House. He spoke at a May 23 debate on the topic organized by the Center for Strategic and International Studies in Washington.

Other nuclear experts argue that the perils of stopping short of a complete upgrade far outweigh the costs. “This is something that we’ve got to decide it’s time for us to invest, and we’ve got to get moving,” C. Robert Kehler, a retired Air Force general and a former commander of the United States Strategic Command, said at the same event.