Secretary of State Rex Tillerson and others dismissed questions about whether the United States is sending a message to North Korea, for example, that undermines any deal that nation might contemplate over its own nuclear program. (Andrew Harnik/AP)

The United States will remain a trustworthy international partner, administration national security aides said Sunday, offering reassurances after allies and members of Congress criticized President Trump for deciding to alter terms for participation in the international nuclear deal with Iran.

Secretary of State Rex Tillerson and others dismissed questions about whether the United States is sending a message to North Korea, for example, that undermines any deal that nation might contemplate over its own nuclear program.

"I think what North Korea should take away from this decision is that the United States will expect a very demanding agreement with North Korea, one that is very binding and achieves the objectives, not just of the United States but the policy objectives of China and other neighbors," Tillerson said in an interview on CNN's "State of the Union."

That shared goal, he said, is to rid the Korean Peninsula of nuclear weapons.

"We intend to be very demanding in that agreement," Tillerson said. "And if we achieve that, then there will be nothing to walk away from."

Trump on Friday set further conditions on U.S. participation in the landmark 2015 nuclear agreement with Iran and five other nations and threatened to walk away if his concerns are not met. His decision threatens but does not undo a signature foreign policy priority of former president Barack Obama that Trump has called "an embarrassment."

European allies have warned that they will not follow suit if the United States abrogates the deal. Although Trump did not kill the agreement outright, as he had frequently threatened to do, the current limbo is not much better from the perspective of several key allies.

German Foreign Minister Sigmar Gabriel said Saturday that Trump's action increases the threat of war near Europe and risks a spillover effect for other conflicts.

"My big concern is that what is happening in Iran or with Iran from the U.S. perspective will not remain an Iranian issue, but many others in the world will consider whether they themselves should acquire nuclear weapons, too, given that such agreements are being destroyed," Gabriel said.

Tillerson said on Sunday, "The issue with the Iran agreement is, it does not achieve the objective. It simply postpones the achievement of that objective. And we feel that that is one of the weaknesses under the agreement, so we're going to stay in. We're going to work with our European partners and allies to see if we can't address these concerns, which are concerns of all of us."

U.N. Secretary Nikki Haley was asked what incentive North Korean leader Kim Jong Un would have to cut a deal now. Such an agreement has long been presumed to require American leadership and backing, since Kim considers Washington his principal enemy. Many U.S. officials believe Kim's rapidly advancing nuclear weapons capability is aimed at preventing a U.S. attack or increasing Kim's leverage in an eventual international negotiation.

"The whole reason we are looking at this Iran agreement is because of North Korea," Haley said on NBC's "Meet The Press."

"What we're saying now with Iran is, don't let it become the next North Korea. So what this says to North Korea is, 'Don't expect us to engage in a bad deal, and also, if at any point we do come up with something, expect us to follow through with it. Expect us to hold you accountable.'"

Trump's national security adviser, H.R. McMaster, said that the president's threat to cancel the Iran deal "set out a marker" for the United States and its allies to fix what he called "a weak deal that is being weakly monitored."

"The president has made clear that he will not permit this deal to provide cover for what we know is a horrible regime to develop a nuclear weapon," McMaster said during an appearance on "Fox News Sunday."

"One of the real problems with this deal is we can't really say with confidence that they're complying," McMaster said, accusing Iran of having "walked up to the line" and "crossed the line several times in terms of the restrictions."

"This is not a trustworthy regime," he said. "So much more comprehensive monitoring is in order."

Asked what incentive Iran has to revisit the deal, McMaster said, "They have to revisit it because otherwise what you do is you just give the Iranians the opportunity to develop a nuclear capability. Their programs can advance and then they can go to industrial scale enrichment of uranium within a very short period of time and then bridge into a weapon, and that is just an unacceptable risk to the world."

McMaster insisted that "the president's not walking away from the deal yet" and wants to see "some real change."

Tillerson said Trump's action on Iran is a "signal" to Congress and the other signers of the agreement, including Iran.

"If we don't see improvement, there is no sense in staying in, and he has every intention of walking out," he said on CBS's "Face The Nation." The action Friday puts the onus on Congress to decide what to do next, including attempting to add provisions that would satisfy U.S. concerns, Tillerson said.

A separate new agreement among the international parties to the deal could "lay alongside" the existing 2015 pact, Tillerson said. He dismissed criticism that Trump's opposition to the Iran deal as it stands would spoil chances for a similar international compact to address North Korea's nuclear program.

"It would be pretty rich" for North Korea to doubt U.S. trustworthiness, Tillerson said with a smile. He said the diplomatic approach he is pursuing "will continue until the first bomb is dropped."

Sen. Lindsey O. Graham (R-S.C.) said Trump is taking the right approach to Iran and a deal the senator said he agrees is deeply flawed.

"The president ran on the idea that this was a bad deal for America, and he won," Graham said.

He said he could support the approach favored by GOP Sens. Bob Corker (Tenn.) and Tom Cotton (Ark.) to add "triggers" for Iranian behavior that would lead to a return of U.S. sanctions suspended under the deal. That would require legislation, and some Democrats say it appears to be only a precursor to an eventual U.S. withdrawal from the deal.

Sen. Chris Murphy (D-Conn.), a member of the Foreign Relations Committee, said it is "an absolute fantasy" to believe that Iran deal negotiations will be reopened.

"The Iranians will not renegotiate it, and neither will the Europeans," Murphy said on "Fox News Sunday." "And so if we were to pull out of this agreement, as the president is threatening, Iran would get everything they want. They would be able to restart their nuclear program, because we would be in violation of the deal. The Europeans would continue to grant them sanctions relief, their economy would continue to grow and they would look like the victim in the situation."

Murphy also said that Iran is complying with the agreement.

"The White House is required to tell Congress if they are not, and they've submitted absolutely no submissions," he said.

Hillary Clinton sharply criticized Trump on Sunday, arguing that his declaration "says America's word is not good."

Appearing on CNN's "Fareed Zakaria GPS," the former Democratic presidential candidate and former secretary of state noted that others in Trump's administration had advised against the move and that Trump was decertifying a deal "in the absence of evidence that Iran is not complying."

"That is bad not just on the merits for this particular situation, but it sends a message across the globe that America's word is not good," Clinton said. "We have different presidents, and this particular president is, I think, upending the kind of trust and credibility of the United States' position and negotiation that is imperative to maintain."

"I know that Iran plays a game of aggressiveness and undermining of our interests and the interests in the region — there's no argument about that," Clinton said. "But my point has been and remains, I would much rather deal with Iran's other bad behavior while not worrying at this moment about their nuclear program getting up and going again. And why on earth would we want two nuclear challenges in Iran and North Korea at the same time?"

Iranian Foreign Minister Mohammad Javad Zarif also said that the United States is becoming less credible as a partner.

Zarif, interviewed on the same CBS program, said Tillerson had not called him to preview Trump's speech Friday, "and I didn't expect him to." Zarif, who led negotiations for Iran, said Trump is undermining global confidence in the United States. "The United States is no longer just unpredictable but unreliable," Zarif said in an interview in Tehran.

Trump's advisers were also asked about turmoil among those advising the president on national security matters.

Tillerson defended his standing within the administration Sunday, saying Trump's social-media pronouncements on major global issues did not undercut the nation's top diplomat.

"I'm not going to deal with that kind of petty stuff," Tillerson said on CNN, rejecting a chance to address reports that he called Trump a "moron" after a national security meeting at the Pentagon in July.

Corker, a close Tillerson ally, also said last week that Trump had "castrated" the secretary of state with tweets undermining his diplomatic efforts over North Korea.

"I checked; I'm fully intact," Tillerson said on CNN.

Tillerson said that his relationship with the president, often described as strained, was just fine. "I call the president Mr. President," Tillerson said. "We have a very open exchange of views."

He ran into similar questions on CBS but gave no sign of irritation or dismay. Tillerson said there is close coordination among the administration's national security agencies and pointed to what he called a productive relationship with China over North Korea policy as an example of how he and the White House work together.

He also put a positive gloss on Trump's unpredictable behavior.

Trump likes to "cause action" with unpredictable or bold statements, Tillerson said.

Haley deflected reports that she and Tillerson are butting heads amid suspicion that she will replace him at Foggy Bottom.

"That's ridiculous," she said on ABC's "This Week."

Paul Kane contributed to this report.