A top Republican lawmaker warned on Sunday that US President Donald Trump's impulsive threats against other countries could set America "on the path to World War III," in comments capping an unusually hostile and public feud between the pair on Twitter.

Bob Corker, who chairs the powerful Senate Foreign Relations Committee, slammed the president for running his office like a "reality show" in a stinging rebuke against a sitting president of his own party that was remarkable even by the volatile standards of the Trump administration.

The public spat between the former allies could also undermine Trump's legislative agenda, with Corker's vote vital to the fate of the Iran nuclear deal and passing tax reform.

"He concerns me," Corker told The New York Times, adding: "He would have to concern anyone who cares about our nation."

The drama erupted in the morning, when Trump tweeted that Corker had decided against running for re-election because he lacked the "guts," and the senator replying that the White House had become an "adult daycare centre."

The highly unusual exchange came seemingly out of nowhere on a rainy Sunday morning that Trump began in the White House before skies cleared and he headed to a nearby golf course.

Corker, a respected moderate who once supported Trump, has emerged in recent months as one of the president's most outspoken Republican critics.

He recently said that only the presence of the generals in Trump's inner circle had kept the White House from descending into "chaos."

Trump made no mention of that remark in his volley of tweets, instead attacking Corker for his decision to return to private life, saying that the Tennessee Republican bore heavy responsibility for what Trump considers the deeply flawed nuclear deal with Iran.

"Senator Bob Corker 'begged' me to endorse him for re-election in Tennessee. I said 'NO' and he dropped out (said he could not win without my endorsement)," Trump wrote in a series of tweets.

"He also wanted to be Secretary of State, I said 'NO THANKS.' He is also largely responsible for the horrendous Iran Deal!"

Trump called Corker a "negative voice" who would "stand in the way of our great agenda."

"Didn't have the guts to run!" the president exclaimed.

Corker for his part disputed the account, telling the Times that Trump had urged him to run again and promised to endorse him if he did.

"I don't know why the president tweets out things that are not true," he said.

The generals and 'chaos'

Trump's tweetstorm was not particularly unusual -- he has rarely been one to calmly endure criticism -- but Corker's blunt rebuttal was strikingly rare, coming from a fellow Republican in such a prominent position.

The senator, perhaps feeling liberated by his decision not to seek re-election, tweeted: "It's a shame the White House has become an adult day care center. Someone obviously missed their shift this morning," hinting that presidential caretakers were not doing their job.

A few hours later, Trump was online once again -- hitting back to suggest Corker's achievements in office have been limited.

"Bob Corker gave us the Iran Deal, & that's about it. We need HealthCare, we need Tax Cuts/Reform, we need people that can get the job done!" he wrote.

The Trump-Corker feud, long in the making, flared up last week after Trump seemed to pull the rug out from Secretary of State Rex Tillerson as the Texan attempted to open lines of dialogue with North Korea.

Trump tweeted that Tillerson was "wasting his time trying to negotiate."

A clearly frustrated Corker told reporters on Wednesday that Tillerson was not getting the support he needed from above.

He then added: "I think Secretary Tillerson, Secretary (of Defense James) Mattis and Chief of Staff (John) Kelly are those people that help separate our country from chaos, and I support them very much."

While he did not mention Trump specifically, the comments seemed clearly aimed at the president.

Corker had supported Trump's presidential campaign. He reportedly was considered as a possible vice presidential running mate to Trump, and later as a potential secretary of state.

But the relationship took a turn for the worse in August. After Trump's much-criticised remarks about the violent protests in Charlottesville, Virginia, Corker said the president had "not yet been able to demonstrate the stability, nor some of the competence, that he needs to demonstrate."