Washington (CNN) President Donald Trump threatened Iran with "obliteration" on Tuesday, saying that an attack on "anything American will be met with great and overwhelming force."

"In some areas, overwhelming will mean obliteration. No more John Kerry & Obama!" the President tweeted.

Speaking in the Oval Office later in the day, Trump said he believes Iran still takes his threats seriously after he canceled a planned military strike and told reporters he does not need an "exit strategy" from the increasingly tense situation.

Trump also said his message to Tehran following the recent exchange of hostile words between the US and Iran -- including Iranian President Hassan Rouhani saying the White House is suffering from a "mental disability" -- was that he wants to negotiate.

"I'll tell you what the message is: When they're ready, they have to let us know," Trump said. "Whatever they want to do, I'm ready."

"You're not going to need an exit strategy. I don't need exit strategies," he added.

Those comments came hours after Trump had responded to Rouhani's comments on Twitter, saying, "Iran's very ignorant and insulting statement, put out today, only shows that they do not understand reality."

"Their leadership spends all of its money on Terror, and little on anything else. The U.S. has not forgotten Iran's use of IED's & EFP's (bombs), which killed 2000 Americans, and wounded many more," he added.

Trump's figures regarding Iranian responsibility for American deaths appeared to be significantly higher than those provided by the State Department and the Pentagon in April, which said that "at least 603 US personnel deaths in Iraq" were the result of attacks by Iran-backed militants between 2003 and 2011.

Rouhani also said that those "in charge of the White House are feeling frustrated" by the state of play in the region, according to the semi-official Tasnim news agency. He added that the US had wrongly expected to "create chaos" in Iran in two to three months, during his speech to senior health officials.

During an unannounced trip to Afghanistan, Secretary of State Mike Pompeo called Rouhani's comments, "a bit immature and childlike."

"But know that the United States will remain steadfast," he added.

Speaking to reporters Sunday, Pompeo said "we're prepared to negotiate with no preconditions," adding that he was "confident that at the very moment they are ready to engage with us, we will be able to begin these conversations."

The secretary of state has previously issued 12 demands for change in Iran before the US will ease its maximum pressure campaign against Iran. The Trump administration has argued that the Joint Comprehensive Plan of Action, the nuclear deal it abandoned in 2018, is inadequate as it doesn't cover Iran's ballistic missiles or regional activities.

Yet, tensions between the US and Iran are now at their highest level in years, coming on the back of last week's downed US drone, but also stretching to 2018 when Trump walked away from the Iranian nuclear deal implemented by his predecessor Barack Obama.

Iran has since threatened to exceed enrichment limits allowed under the nuclear deal and is blamed by the US for attacks on two tankers in the vital Gulf of Oman waterway earlier this month. Iran denied responsibility for the ship attacks.

The situation continued to escalate when Trump threatened airstrikes on Iran last week -- calling them off just minutes before they were due to begin.

On Tuesday, his national security adviser, John Bolton, continued the administration's tough rhetoric.

Bolton referred to Iran as a "radical regime" that supports "violent provocations abroad," ahead of a trilateral meeting with his Israeli and Russian counterparts in Jerusalem.

But he also added that Trump had "held the door open to real negotiations."

"All that Iran needs to do is walk through that open door," said Bolton, known for being one of the administration's most hawkish advisers on Iran.

Bolton said later Tuesday that Trump called him prior to sending tweets threatening Iran and said the President asked him to "get the message out" that Iran, in Trump's words, will face "great and overwhelming force" if it attacks "anything American."

That message, however, has not been well received by at least one US ally in the region -- Iraq, whose President Barham Salih told CNN that under no circumstances will the US be allowed to launch attacks on neighboring Iran from its bases in his country.

"We do not want our territory to be a staging post for any hostile action against any of our neighbors, including Iran," Salih said in an interview with CNN's Christiane Amanpour in London on Tuesday.

"This is definitely not part of the agreement between the Iraqi government and the United States."