Last month, President Donald Trump said he was willing to meet with Iran's leaders "anytime they want" and without preconditions. Bolton reiterated that invitation and said that "if the ayatollahs want to get out from under the squeeze, they should come and sit down."
"They should not underestimate our determination that we're going to put pressure on them until they give up their pursuit of nuclear weapons and all the other activities that I mentioned," Bolton said. "That, we are very serious about."
Iran explicitly agreed not to pursue nuclear weapons in the 2015 agreement. US intelligence agencies have concluded that Iran halted a nuclear weapons programme in 2003 and did not have one at the time of the accord.
Congressional reaction to the restoration of sanctions split largely along party lines.
Republicans hailed the move as a much-needed step towards putting pressure on Iran to end its support for militants in the region, halt its ballistic missile programme and agree to modify sunset provisions in the nuclear accord.
"Under these new sanctions businesses throughout the world will have to choose to do business with the American economy or Iranian economy? You can't do both," Senator Lindsey Graham tweeted.
"Make Iran Great Again," he added, in a new twist on Trump's campaign slogan. "Dump the Ayatollah!"
Democrats argued that the move would make the world more dangerous, not safer.
Senator Richard Durbin, who was instrumental in shepherding the original Iran deal through Congress three years ago, said that he remains opposed to the Trump Administration's "irrational hostility" to the agreement.
"Today's actions, yet again, put the US in violation of this deal. It risks reopening a resolved conflict, and will divide us further from our European allies. President Trump's foreign policy is a dangerous gamble with nuclear weapons," Durbin said.