"We do need the support of our allies, our European allies and others, to make the case as well to Iran that this deal really needs to be revisited," Tillerson told the Fox News television channel.
The US diplomat-in-chief spoke on the eve of his first meeting with Iranian Foreign Minister Mohammad Javad Zarif and foreign ministers from the five other signatories of the 2015 nuclear accord: Britain, China, France, Germany and Russia.
In his first address to the UN General Assembly, President Donald Trump signaled he was ready to kill the nuclear deal, dubbing it "an embarrassment to the United States."
Under the nuclear deal, Iran surrendered much of its enriched uranium, dismantled a reactor and submitted nuclear sites to UN inspection, while Washington and Europe lifted some sanctions.
French Foreign Minister Jean-Yves Le Drian this week floated the idea of opening talks on provisions that set 2025 as the date for some restrictions on Iranian uranium enrichment to be lifted.
Tillerson said that "sunset provision" was "the most glaring flaw" and drew a parallel with North Korea where an agreement on dismantling its nuclear program collapsed in 2002.
"It's not a stiff enough agreement. It doesn't slow their program enough," said Tillerson.
"We can almost start the countdown clock as to when they will resume their nuclear weapons capability."