Sen. Bob Corker of Tennessee surrounded by reporters on Capitol Hill on July 27. AP Photo/J. Scott Applewhite

Republican Sen. Bob Corker stepped up his criticism of President Donald Trump late Sunday, saying the president was running the country like a "reality show" and, through his rhetoric, could set the US "on the path to World War III."

The comments, appearing in an interview with The New York Times, were an escalation of the testy back-and-forth between Corker and Trump that began last week after the Tennessee senator announced he would not seek reelection in 2018.

"He concerns me," Corker said, according to The Times. "He would have to concern anyone who cares about our nation."

Corker also suggested that Trump may not completely grasp the power of the presidency.

"I don't think he appreciates that when the president of the United States speaks and says the things that he does, the impact that it has around the world, especially in the region that he's addressing," he told The Times. "And so, yeah, it's concerning to me."

Earlier Sunday, Trump blasted Corker in an early-morning tweetstorm, claiming he denied the senator his endorsement and rejected him as a candidate for secretary of state. Trump also blamed Corker for what he called the "horrible Iran Deal" and said the senator "didn't have the guts" to run for reelection.

Corker quickly fired back, tweeting that it was a "shame the White House has become an adult day care center." He also denied the president's claims in his interview with The Times, saying it was Trump who encouraged Corker to run for reelection and preemptively offered his endorsement, only for Corker to say he had decided not to run again.

"I don't know why the president tweets out things that are not true," he told The Times. "You know he does it, everyone knows he does it, but he does."

The public quarrel between Corker and Trump began Wednesday when Corker, who chairs the Senate Foreign Relations Committee, sharply criticized the president for undermining Secretary of State Rex Tillerson's efforts to establish diplomatic channels with North Korea amid heightening nuclear tensions. Corker described Tillerson as being among "those people that help separate our country from chaos."

Corker was also vocally critical of the Republican tax plan, saying last Monday that it could balloon the federal deficit.

With Republicans holding a slim 52-seat majority in the Senate, Trump can ill afford to lose Corker's vote on his proposed tax overhaul.