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.

More: Saudi Arabia Promises to Match Iran in Nuclear Capability