US Vice President JD Vance said on Sunday that there had been "a number of good conversations with the Iranians" after officials from the Islamic regime met with Qatari mediators in Doha.
Qatar added that there was an agreement to avoid escalation in the region and the gulf, and that both the US and Iran were committed to the mediation process.
Mediators are working to bridge gaps between Tehran and Washington, as overnight US strikes on Qeshm Island and Iran's downing of a US drone threaten to complicate negotiations.
"An American nuclear attack on Iran would set a new precedent," Carlson claimed in a post on X/Twitter a day before the US carried out "defensive" strikes on Qeshm Island.
US officials said the strikes were defensive measures against Iranians using surface-to-air missiles against US assets, and did not appear to signal a resumption of major combat operations.
Trump replied "no" when asked on Monday if he was considering resuming strikes on Iran, despite Prime Minister Benjamin Netanyahu's insistence on doing so.
China's Foreign Minister Wang Yi and the Russian Foreign Ministry both urged all parties to the conflict to focus on political solutions rather than resorting to arms again.
Trump confirmed that he had spoken to Netanyahu while speaking to reporters on Sunday, and that they had discussed Iran but declined to share any further details.
"One of the vessels is a supertanker; it is currently engulfed in flames following a powerful explosion," the Iranian Revolutionary Guard Corps (IRGC) Navy spokesperson said.
Trump and his non-military advisors were also worried about the impact of an additional round with Iran, which is unpopular in the US, on the Republican Party's chances in the midterm elections.
US-Iran negotiations show some progress through Qatari mediation, but major disagreements over Tehran's nuclear program threaten a potential deal.
