The Trump administration is claiming some early success in its quest to cripple Iran with sanctions in order to drag it back to negotiations and change its behavior, even as it softens its demand on U.S. allies to reduce their Iranian oil imports to zero.

Since President Donald Trump withdrew the U.S. from the Iran deal, his administration has been pushing for allies to stop importing Iranian oil before sanctions snap back into place in November. But some key countries, including India and Turkey, have already said they will not do so.

While a senior State Department official said Monday that they are "not looking to grant licenses or waivers," he did say the administration will work with countries on a "case-by-case basis" while their "diplomacy "has been focused around mostly consultations with Europe, France, and Germany," not other countries such as China or Russia that would be more difficult to persuade.

"We will not hesitate to take action when we see sanctionable activity," said Brian Hook, the department's Policy Planning Director and the former lead negotiator for the U.S. who met with European allies for months to reach a side agreement that kept the U.S. in the Iran nuclear deal but addressed Trump's concerns with it. Despite nearly reaching a deal with the Germans, French, and British, President Trump dismissed their progress and withdrew the U.S. from the landmark nuclear accord anyway.

In the face of the damage that did to U.S.-European relations, Hook and teams from the State Department and Treasury Department have been traveling to European and East Asian countries to shore up support for this new pressure campaign and try to persuade countries to reduce oil imports. This week, he and Under Secretary of the Treasury for Terrorism and Financial Intelligence Sigal Mandelker will travel to Gulf countries to begin talks there.

"Our focus is on getting as many countries importing Iranian crude down to zero as soon as possible," Hook said – a slight step back from what a senior State Department official who briefed reporters last week said, that all allies must be at zero imports by November 4. That official, who spoke without attribution, said, "Without question, they should be reducing... They should be preparing now to go to zero."

Iran has said it will fight back against US efforts to squeeze Iranian oil exports, accusing countries that fill the gap in oil markets of "a big treachery to the Iranian nation and the world community," according to Iran's First Vice President Eshagh Jahangiri.

After Trump's withdrawal from the nuclear deal, the first round of sanctions will be reimposed on August 6, targeting Iran's gold, other metals like aluminum and steel, and automobile industry. One-hundred and eighty days later, on November 4, sanctions will snap back on Iran's energy sector, all oil-related transactions, and transactions with the Central Bank of Iran.

Hook seemed to celebrate the collapsing Iranian economy, telling reporters Monday that their efforts had led to more than 50 international companies announcing they will leave the Iranian market, although the State Department is not providing that list.

"Foreign direct investment is falling, and the rial hit an all-time high against the dollar last week," he added, blaming the Iranian regime's "violent misadventures abroad" and saying its economy is "too distorted by corruption and the [Iranian Revolutionary Guard Corps'] pervasive presence in most key sectors."

The collapsing economy has sparked new protests in Iran, with gunfire erupting Sunday morning as Iranian security forces confronted protesters demonstrating over water scarcity in the country's south. According to the Associated Press, peaceful protests began in Khorramshahr, Abadan and other areas of Iran's oil-rich Khuzestan province on Friday, but over the weekend, protesters began throwing stones and confronting security forces in at least one city.

There were also three days of demonstrations last week in Tehran, with groups confronting police outside parliament and rallies temporarily shutting down the city's Grand Bazaar, according to the AP.

While the administration's goal is to create maximum pressure on Iran "until the regime changes its destabilizing policies," it's unclear if that's where the sharp economic pain will lead, or whether it could embolden hardliners, inflame anti-American sentiments, or bring the country to some unknown chaos that could be worse.

Under the Obama administration, oil sanctions were targeted so that countries had to continually reduce their Iranian oil imports every 180 days or face U.S. sanctions, but the Trump administration's tougher line could mean a sudden collapse of the oil-dependent Iranian economy or a sharp spike in oil prices worldwide.

Hook said Monday the U.S. is "working to minimize disruptions to the global market, but we are confident that there is sufficient global spare oil production capacity." In particular, the administration points to a recent announcement by Saudi Arabia to increase oil production, with President Trump tweeting Saturday that the country's leader King Salman agreed to increase oil production by 2 million barrels. The Saudis have not confirmed if that's the case.

The U.S. does not seek to overthrow the Iranian regime, Hook reiterated Monday, but change its behavior instead.

But two key administration allies – Newt Gingrich and Rudy Giuliani – joined a rally in Paris over the weekend organized by an exiled opposition group that calls for the regime's violent overthrow. Speaking at an event organized by the Mojahedin-e Khalq, or MEK, Giuliani, who is Trump's personal lawyer in the Russia investigation, called for ramping up the pressure on the regime.

"This government is about to collapse, and this is the time to turn on the pressure," Giuliani told the crowd, according to Bloomberg News.

The State Department has insisted on multiple occasions that Giuliani does not speak on behalf of the administration on foreign policy issues.