(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.

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