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That, of course, was the central dilemma facing the Obama administration making its first, secret approach to Iran six years ago. At first, Mr. Obama’s aides insisted Iran would have to give up everything, but that the Tehran government could produce no material that might ultimately be diverted to a bomb.

Eventually, American negotiators concluded after years of running into walls that it would be better to leave Iran with a face-saving token capability for 15 years, and vigorous international inspections, than walk away with no agreement and the real prospect of war.

Many of Mr. Obama’s critics, including some Democrats, have said the negotiators gave up too much. During the 2016 presidential campaign, Mr. Trump quickly honed in on the nuclear agreement’s most glaring weakness: After 15 years, the Iranians could resume unlimited fuel production.

Mr. Obama’s essential bet was that in 15 years Iran will have different leadership, perhaps more interested in integrating with the world than keeping a bomb-making capability. So he brought into the negotiation a nuclear scientist in his cabinet, Energy Secretary Ernest Moniz.

Mr. Moniz, the former head of the M.I.T. nuclear physics lab, sat for months with his Iranian counterpart, who had done his graduate studies at M.I.T. They bonded, and emerged with an agreement that Energy Department scientists certified would assure Iran would need a year or more to “break out” and manufacture the fuel needed for a nuclear weapon — until the 15-year clock ran out.

Now Mr. Trump’s negotiators have decided they need the same thing — but it must be permanent.

The schedule that Mr. Rouhani announced to his nation last week would put Iran back on the path of nuclear fuel production. Sooner or later, it would cross that one-year threshold. Iran has never enriched at the level of purity needed to produce a weapon, inspectors say, but they have come close.

“If you want to keep Iran more than a year away from the capability to build a bomb, the way to do it is to go back into the deal,” said Jake Sullivan, a former Obama administration national security official who helped open the negotiations with Tehran. “Because that’s exactly what the deal does.”