By Barbara Starr, CNN Pentagon Correspondent

(CNN) -- A month after sending an aircraft carrier to the Middle East in reaction to intelligence the US claims it had showing Iran was preparing to attack US troops, military tensions appear to be easing, according to several US officials.

"It seems tensions have dropped some, but we are still watching very closely, we haven't relaxed, we remain vigilant," one defense official with direct knowledge of the situation told CNN.

While officials have not publicly shown any of the intelligence that they said led to the deployment of the carrier, as well as aircraft and air defense missiles, new information has emerged underscoring why the US was so concerned over the weekend beginning Friday, May 3.

Several officials tell CNN that even as initial messages were sent to Iran through an unidentified third party on that day, the Pentagon had intelligence that the regime was not taking the US warning seriously. Then, by Sunday, the US took the next step by making public it was dispatching military forces. CNN has not seen the messages.

President Donald Trump did little to dampen the tensions when he said this week there is "always a chance" the US might take military action against Iran.

"There's always a chance. Do I want to? No, I'd rather not but there's always a chance" Trump said in an interview with ITV's Piers Morgan that was taped Tuesday. Trump did reiterate that he is willing to talk to Iran's President Hassan Rouhani.

"The only thing is we can't let them have nuclear weapons," Trump said, after making clear he is willing to talk.

'Maximum pressure'

Tensions with Iran have risen following the Trump administration's decision to withdraw from the Iran nuclear agreement that was meant to provide Iran with sanctions relief in exchange for curbs on its nuclear program. The US has also doubled down on its "maximum pressure" campaign, which has seen tougher sanctions on Tehran introduced in recent months.

Officials have not said exactly how they knew Iranian officials were not taking the US warning seriously, but Gen. Joseph Dunford, chairman of the Joint Chiefs, dropped a possible hint during a recent appearance at the Brookings Institution.

After the apparent threat of a plot to attack emerged on May 3, Dunford told the audience at Brookings that "we also saw in the intelligence that perhaps there was a question about the will and capability of the United States to respond."

Dunford's office declined to comment specifically on what he was referring to. But several other officials pointed out there were at least four separate streams of intelligence, some of which included communications intercepts and human intelligence. It is that type of intelligence that could provide information about the views and intentions of Iranian officials.

Several officials also say the intelligence remained classified at an unusually high level, typical of electronic intercept information and human intelligence because it reveals sources and methods of collecting information. That was combined with overhead imagery from satellites and aircraft which show military capabilities on the ground but do not give an insight into the intentions of US adversaries.

Dunford also offered additional details about how the threat picture emerged that first weekend in May. "In the last weekend of April, I began to see more clearly things that I had been picking up on over a period of months." Dunford said he remembers clearly that on May 3, "multiple threat streams that were all perhaps coming together in time." It was then that the plan to send forces in the region quickly took shape and an announcement from national security adviser John Bolton was made about the deployment.

Rhetoric isn't weakening

The rhetoric from both sides, however, isn't weakening. On Tuesday, Iranian media reports said the Supreme Leader Ayatollah Khamenei said "Americans have to stay away" during ceremonies commemorating the 30th anniversary of the death of Imam Khomeini, founder of the Islamic Republic.

US officials speaking on background strongly believe that the threats from Iran to attack US interests and the continuing rhetoric are a reflection of the regime's need to appear strong in the face of sanctions on its oil industry and other sectors such as industrial metals.

For now, Iran appears to have removed missiles from two small ships, known as dhows, which were seen by the US in the opening days of the standoff. Iran had defended outfitting the boats at a deterrence measure against the US in the region.

But one US official with direct knowledge of the Iranian threat says the boats were believed to be "a first strike weapon" by Iran that was 'non survivable." The US believes if the Iranians had fired the missiles, they would have quickly destroyed the boats to eliminate evidence.

Several officials have said the US intelligence, which CNN has not seen, also showed that Iran's Quds Force commander Qasem Soleimani was deeply involved in the potential attack plans the US saw that included a potential Iran attack on US troops in Iraq, a maritime attack in the Persian Gulf, and strikes against Saudi and UAE oil facilities by Iranian units and Iranian backed Houthi fighters.