Speaking alongside Israeli Prime Minister Benjamin Netanyahu in Jerusalem, Bolton said no one has granted Iran a "hunting license in the Middle East."
"As President Trump said on Friday our military is rebuilt, new and ready to go," Bolton said.
The comments come days after Trump confirmed that he was "cocked and loaded" to strike Iranian targets but deemed the loss of life would be disproportionate to the downing of an unmanned U.S. drone earlier last week.
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"Iran's continued pursuit of nuclear weapons, its threats to exceed the limits set in the failed Iran nuclear deal in the coming days ... are not signs of a nation seeking peace," Bolton told reporters.
"Sanctions are biting, and more added last night," he said. "Iran can never have nuclear weapons — not against the U.S.A. and not against the world."
Netanyahu has himself been a vocal critic of Iran over the years. Israel considers Iran to be their greatest threat, but the prime minister and Trump ally has remained uncharacteristically quiet throughout the current crisis.
Trump said Friday that he doesn't want war with Iran, but if it comes there will be "obliteration like you've never seen before."
"But I'm not looking to do that," the president added in his interview with NBC. He said there were no pre-conditions for U.S. talks with Tehran.
The drone strike and the aborted decision to strike Iranian targets followed weeks of rising tension and attacks on two oil tankers in the Gulf last week.
The developments are underpinned by a deep mutual mistrust between the nations, as well as U.S. criticism of Iran's nuclear and ballistic missile activities and support for proxies in various Middle East conflicts.
Trump pulled the U.S. out of a joint nuclear deal with Iran last year, making good on a campaign promise.
Since shelving the strike, Trump has appeared to be more focused on continuing to squeeze Iran financially through sanctions.
Iran cannot have Nuclear Weapons! Under the terrible Obama plan, they would have been on their way to Nuclear in a short number of years, and existing verification is not acceptable. We are putting major additional Sanctions on Iran on Monday. I look forward to the day that.....