Trump also thanked Iran for the "wise" decision not to shoot down an American military plane with a crew of 38 aboard. Trump said Iran had a manned plane "in their sights" but instead hit the remotely-piloted surveillance plane, "and that's something we really appreciate."
Conservative critics excoriated the Obama Administration when then-Secretary of State John Kerry publicly thanked Iran for releasing a group of American sailors who had apparently strayed into Iranian waters in 2016.
The President spoke as he left the White House for a weekend getaway at Camp David, where he said he will hold meetings on Iran. He suggested that some advisers would join by video or telephone conference. National security adviser John Bolton is overseas.
Trump said additional US sanctions will soon be applied to Iran, but did not specify the type. His Administration has already applied stringent economic sanctions because of the country's alleged support for terrorism, and Bolton and others have made no secret that a goal is to cripple Tehran's oil-dependent economy.
Bolton, whom Trump called a "hawk" today, had called for regime change in Iran before joining the Administration. Trump rejects that path, but said that he likes hearing a range of views.
"Ultimately I make the decision. The only one that matters is me. I listen to everybody. I want people on both sides," he said.
The President's unusually friendly tone toward the chief US adversary in the Middle East included an open invitation to Iranian leadership to "start all over" with negotiations on its nuclear programme following his decision to withdraw from the 2015 multinational pact negotiated under Barack Obama.
"Iran wants to become a wealthy nation again. Let's make Iran great again. Does that make sense? Make Iran great again, okay with me," Trump said. "But they're never gonna do it if they think in five or six years, they're gonna have a nuclear weapon."
He continued: "We are not going to have Iran have a nuclear weapon. And when they agree to that, they are going to have a wealthy country, and they are going to be so happy, and I'm going to be their best friend. I hope that happens, but it may not."
Trump said the risk of casualties was the main reason not to carry out the airstrikes. The Iranian calculation to avoid targeting the manned US plane appears to have been a major factor in his decision.
"I'm getting a lot of praise for what I did," he said. "My expression is, We have plenty of time."
Israeli Prime Minister Benjamin Netanyahu has said Iran could race for a nuclear weapon in a matter of months, although that possibility seems remote even after announcement last week that Iran would resume some uranium enrichment it had suspended under the 2015 agreement.
Both Trump and Netanyahu call that agreement a mistake and say Obama gave Iran a sweetheart deal. Trump repeated that criticism today, but also warmed to the idea that he might make a new deal and be called a peacemaker in the process.
"We'll start all over and we could have a deal very quickly if they want to do it," Trump said. "It's up to them."
"Everyone was saying I'm a warmonger and now they're saying I'm a dove, and I say I'm neither. I didn't like the idea of them unknowingly shooting down an unmanned drone and we killing 150 people," Trump said.
Tensions have spiked in the Gulf region in recent weeks following a string of assaults on commercial vessels there.
The Trump Administration has blamed Iran for the attacks, which damaged six ships in the waters off the Emirati port of Fujairah. Iran has denied involvement.
In remarks today, the aerospace commander of Iran's Islamic Revolutionary Guard Corps said that the violation of Iran's airspace could have been "a mistake by an American general."
His comments followed Trump's assertion that the drone may have been shot down "unintentionally" by Iran.
Iran said that it summoned the charge d'affaires of the United Arab Emirates over "the violation of the Iranian airspace by an American spy drone that had taken off from the Arab state," the Tasnim news agency reported.
US allies in the Gulf have refrained from directly blaming Iran for the recent attacks on commercial tankers.
