Hoping to avoid that, European diplomats have signaled that they might support an addendum to the 2015 agreement to enforce new limits on Iran’s intercontinental ballistic missile development and testing.
At the center of the negotiations sits Mr. Hook, whom European diplomats believe may represent be the best shot at keeping the nuclear deal alive.
Over meals of schnitzel and schnapps in Vienna, his team sought to bridge the gap between European concerns and Mr. Trump’s demands. It is a precarious balance for the earnest Midwestern political junkie who served as a senior policy adviser to Secretary of State Rex W. Tillerson — before the top American diplomat was fired via presidential Twitter post last week, three days before the talks opened here.
“We are taking things one week at a time,” Mr. Hook, the only aide close to Mr. Tillerson who survived the State Department purge, told reporters on Friday of talks with his European counterparts. “We are having very good discussions.”
European diplomats agreed. Envoys from Britain, France, Germany and policy officials representing the European Union helped negotiate the original deal, in part to improve security for their respective countries. But they share the Trump administration’s concerns about Iran’s activities in Syria, Yemen and elsewhere in the Middle East, which they said should be discouraged.
At least some European officials left Vienna believing that the United States still wants the nuclear accord to survive.
“All parties recommitted to the full implementation of the agreement,” Federica Mogherini, the European Union’s foreign policy chief, told reporters on Monday in Brussels. “It’s, for us, a matter of security for Europe and for the rest of the world.”
Abbas Araghchi, Iran’s deputy foreign minister, also said the American delegation that Mr. Hook led in Vienna “indicated that they are committed.”
Mr. Araghchi added, “As long as there is no new instruction out there, or no new decision at the top level.”
But the clock is ticking and the odds are long.
Mr. Trump has called the deal “the worst ever.” Mike Pompeo, whom the White House has nominated as the next secretary of state, has said he wants it scrapped. And Senator Bob Corker, Republican of Tennessee and the chairman of the Senate Foreign Relations Committee, predicted Sunday that Mr. Trump would soon cast it aside.
For weeks, under Mr. Tillerson’s direction, Mr. Hook has traveled to European capitals to convince his counterparts that the proposed addendum might save the entire Iran deal.
Back in Washington, he also helped formulate much of the Trump administration’s foreign policy and wrote many of Mr. Tillerson’s most important speeches, including a proposal that India join an alliance with the United States, Japan and Australia — a proposal the allies so far have politely dismissed.
But Mr. Hook has never been known in global policy circles for his in-depth knowledge of foreign governments and cultures or for his grand strategic ideas. He has largely relied on sharp political instincts and easy mannerisms to foster American diplomacy abroad.
The son of a banker and grandson of a newspaper publisher, Mr. Hook was raised in one of the most prominent families in Bettendorf, Iowa. He grew up so enamored by the conservative intellectual William F. Buckley’s command of language that he covered the headboard of the bed in his college dormitory room with sticky notes on which he had written obscure English words.
After graduating with a bachelor’s degree in marketing from the University of St. Thomas in Minnesota, Mr. Hook worked for a Republican congressman from Iowa and Gov. Terry E. Branstad of Iowa, also a Republican, before studying and practicing law. He returned to government during President George W. Bush’s administration, working on Iran issues as a political appointee at the State Department and rising to the level of assistant secretary of state.
When President Barack Obama was in office, Mr. Hook helped found the John Hay Initiative, a volunteer network of hundreds of conservative foreign policy, defense and intelligence experts. It represented the heart of the Republican establishment, and many of its members joined the “Never Trump” bandwagon during the 2016 presidential campaign.
Those members disqualified themselves from serving in the Trump administration, but Mr. Hook did not sign on. So when Mr. Tillerson called, Mr. Hook did not hesitate to rejoin the government.
“In campaigns, no policy group creates the kinds of divisive, bitter drama that foreign policy experts bring about. Not education, not economics, nothing else comes close,” said Matt Rhoades, who managed the campaign of Mitt Romney, the Republican presidential nominee in 2012. Mr. Hook served on that campaign as an adviser.
“But Brian brought together hundreds of these people and got them collaborating,” Mr. Rhoades said.
It is precisely this quality that European negotiators hope Mr. Hook can employ for the Iran deal. Representatives for the United States and Europe have mostly agreed on a set of penalties they would impose if Iran develops an intercontinental ballistic missile — a weapon that the Western negotiators agree is useful only to carry nuclear payloads.
But they remain far apart on how to deal with provisions in the accord that currently allow Iran to resume some civil nuclear activities in 2025 and 2030. Washington wants to shut down those activities permanently; the Europeans believe this would renege the terms of the deal. Bridging this disagreement — over what is known as the deal’s sunset provisions — is Mr. Hook’s top goal in his continuing talks.
Mark Dubowitz, who is advising the Trump administration on how to harden the accord, said there was no room for negotiation on this issue.
“On sunsets, I think there can be no compromise,” said Mr. Dubowitz, the chief executive of the Foundation for Defense of Democracies. “They have to be eliminated.”
Even if the United States and Europe find a middle ground to keep the agreement in place — but extend it with new conditions — there is no guarantee that Iran, Russia and China will approve it, as they did the original deal.
President Hassan Rouhani of Iran has ruled out revisiting its terms, warning that Tehran might resume uranium enrichment if the United States withdraws from the accord, scuttling it for the other world powers.
American officials in Vienna for the talks pointed to the Western sanctions that crippled Iran’s economy during the Obama administration as proof that it is the United States and Europe — and not Russia and China — that can force Tehran to comply with the new demands.
European diplomats who were there disagreed. They worried that Iran could worsen an already unstable situation in the Middle East were it to restart its nuclear program now, relying in part on trade with Russia, China and countries that did not negotiate the accord.
Still, the seriousness of the negotiations among the parties has fostered hope. A crucial moment will come in late April, when President Emmanuel Macron of France visits Washington.
For Mr. Hook, the negotiations could be either a capstone or an end to his tenure in the Trump administration.
On Tuesday, the president sounded an ominous tone in chastising Iran for behavior that he described as inappropriate.
“A lot of bad things are happening in Iran,” Mr. Trump said, sitting in the Oval Office next to Crown Prince Mohammed bin Salman of Saudi Arabia, one of Iran’s regional rivals. “The deal is coming up in one month, and you will see what happens.”