WASHINGTON — When President Obama began making the case for a deal with Iran that would delay its ability to assemble an atomic weapon, his first argument was that a nuclear-armed Iran would set off a “free-for-all” of proliferation in the Arab world. “It is almost certain that other players in the region would feel it necessary to get their own nuclear weapons,” he said in 2012.

Now, as he gathered Arab leaders over dinner at the White House on Wednesday and prepared to meet with them at Camp David on Thursday, he faced a perverse consequence: Saudi Arabia and many of the smaller Arab states are now vowing to match whatever nuclear enrichment capability Iran is permitted to retain.

“We can’t sit back and be nowhere as Iran is allowed to retain much of its capability and amass its research,” one of the Arab leaders preparing to meet Mr. Obama said on Monday, declining to be named until he made his case directly to the president. Prince Turki bin Faisal, the 70-year-old former Saudi intelligence chief, has been touring the world with the same message.

“Whatever the Iranians have, we will have, too,” he said at a recent conference in Seoul, South Korea.

For a president who came to office vowing to move toward the elimination of nuclear weapons, the Iran deal has presented a new dilemma. If the agreement is sealed successfully next month — still far from guaranteed — Mr. Obama will be able to claim to have bought another decade, maybe longer, before Iran can credibly threaten to have a nuclear weapon.

But by leaving 5,000 centrifuges and a growing research and development program in place — the features of the proposed deal that Israel and the Arab states oppose virulently — Mr. Obama is essentially recognizing Iran’s right to continue enrichment of uranium, one of the two pathways to a nuclear weapon. Leaders of the Sunni Arab states are arguing that if Iran goes down that road, Washington cannot credibly argue they should not follow down the same one, even if their technological abilities are years behind Iran’s.

“With or without a deal, there will be pressure for nuclear proliferation in the Middle East,” said Gary Samore, Mr. Obama’s top nuclear adviser during the first term and now the executive director of the Belfer Center for Science and International Affairs at Harvard. “The question is one of capabilities. How would the Saudis do this without help from the outside?”

In fact, the Arab states may find it is not as easy as it sounds. The members of the Nuclear Suppliers Group, a loose affiliation of nations that make the crucial components for nuclear energy and, by extension, weapons projects, have a long list of components they will not ship to the Middle East. For the Saudis, and other Arab states, that leaves only North Korea and Pakistan, two countries that appear to have mastered nuclear enrichment, as possible sources.

It is doubtful that any of the American allies being hosted by Mr. Obama this week would turn to North Korea, although it supplied Syria with the components of a nuclear reactor that Israel destroyed in 2007.

Pakistan is another story. The Saudis have a natural if unacknowledged claim on the technology: They financed much of the work done by A.Q. Khan, a Pakistani nuclear scientist who ended up peddling his nuclear wares abroad. It is widely presumed that Pakistan would provide Saudi Arabia with the technology, if not a weapon itself.

The Arab leader interviewed on Monday said that countries in the Gulf Cooperation Council, all to be represented at the Camp David meeting, had discussed a collective program of their own — couched, as Iran’s is, as a peaceful effort to develop nuclear energy. The United Arab Emirates signed a deal with the United States several years ago to build nuclear power plants, but it is prohibited under that plan from enriching its own uranium.

Over the last decade, the Saudi government has financed nuclear research projects but there is no evidence that it has ever tried to build or buy facilities of the kind Iran has assembled to master the fuel cycle, the independent production of the makings of a weapon.

Still, the Saudis have given the subject of nuclear armament more than passing thought. In the 1980s they bought a type of Chinese missile, called a DF-3, that could be used effectively only to deliver a nuclear weapon because the missiles were too large and inaccurate for any other purpose. American officials, led by Robert M. Gates, then the director of the C.I.A., protested. There is no evidence the Saudis ever obtained warheads to fit atop the missiles.

Mr. Obama met with Saudi princes in the Oval Office on Wednesday — Crown Prince Mohammed bin Nayef and Deputy Crown Prince Mohammed bin Salman — who will most likely moderate their criticisms of his administration while talking directly to the president. Mr. Obama is expected to offer them and the other Arab states some security assurances, although not as explicit or legally binding as the kind that protect American treaty allies, from NATO to Japan to South Korea.

But Mr. Obama will have a difficult time overcoming the deep suspicions that the Saudis, and other Arab leaders, harbor about the Iran deal. Several of them have said that the critical problem with the tentative agreements, as described by the White House and Secretary of State John Kerry, is that they assure nothing on a permanent basis.

Prince Turki, while in Seoul, went further. “He did go behind the backs of the traditional allies of the U.S. to strike the deal,” he said of Mr. Obama during a presentation to the Asan Institute for Policy Studies, a South Korean research organization.

Although “the small print of the deal is still unknown,” he added, it “opens the door to nuclear proliferation, not closes it, as was the initial intention.”

Prince Turki argued that the United States was making a “pivot to Iran” that was ill advised, and that the United States failed to learn from North Korea’s violations of its nuclear deals. “We were America’s best friend in the Arab world for 50 years,” he said, using the past tense.

This article was written by David E. Sanger for The New York Times on May 13, 2015. David E. Sanger is chief Washington correspondent of The New York Times. Mr. Sanger has reported from New York, Tokyo and Washington, covering a wide variety of issues surrounding foreign policy, globalization, nuclear proliferation and Asian affairs.