"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.
US 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.
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 Islamic Republic 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.
In an opinion piece published in The Washington Post Iran's Ambassador to the UN Majid Takht-Ravanchi wrote, "The recent dispatching of a US naval armada to the Persian Gulf is a response to the fake intelligence, supported not by members of Congress or US allies."
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.
Tensions with Iran have risen following the Trump administration's decision to withdraw from the Iran nuclear agreement. The US has also doubled down on its "maximum pressure" campaign, which has seen tougher sanctions on Tehran introduced in recent months.