Updated at 4:30 p.m. ET

A leading Republican senator told reporters on Thursday that President Trump "has not yet been able to demonstrate the stability, nor some of the competence that he needs to demonstrate in order to be successful."

Tennessee Sen. Bob Corker was at the Rotary Club of Chattanooga and spoke to local reporters there. In video posted by Chloe Morrison of Nooga.com, Corker added, "And we need for him to be successful. Our nation needs for him to be successful."

Referring to the president's response to the violence that came with white supremacist demonstrations in Charlottesville, Va., last weekend, Corker said, "He also recently has not demonstrated that he understands the character of this nation. He has not demonstrated that he understands what has made this nation great and what it is today."

Corker also warned that without that, "our nation is going to go through great peril" and called for "radical change" at the White House.

Trump said at an impromptu news conference on Tuesday at Trump Tower in New York City that there was blame on "both sides" for the violence in Charlottesville between white supremacists and counterprotesters and that there were "very fine people" on both sides, including some who the president said were just there to peacefully object to the removal of a Confederate statue.

Those statements came after Trump's initial response in the hours after clashes resulted in the death of Heather Heyer, a 32-year-old woman killed when a car slammed into a group of demonstrators on Saturday. In those first comments, Trump said, "We condemn in the strongest possible terms this egregious display of hatred, bigotry and violence," but then added, "on many sides."

He was roundly criticized for not focusing his condemnation on the white supremacists, nor calling out such hate groups by name. He changed course in remarks at the White House on Monday, where he said, "Racism is evil and those who cause violence in its name are criminals and thugs, including the KKK, neo-Nazis, white supremacists and other hate groups that are repugnant to everything we hold dear as Americans."

Trump on Tuesday reverted to his tone from Saturday, which critics say created a moral equivalence between neo-Nazis and those marching against them, in his remarks at Trump Tower. That invited a new and more intense backlash. Trump was also praised by former Ku Klux Klan leader David Duke. Trump has continued to blast the media over coverage of his remarks, tweeting on Thursday, "They totally misrepresent what I say about hate, bigotry etc."

An NPR/PBS NewsHour/Marist poll conducted this week showed that a majority of Americans believe the president's response to the violence in Charlottesville has not been strong enough.

Corker is a respected GOP voice on Capitol Hill, serving as chairman of the Senate Foreign Relations Committee. He was said to be on the shortlist of possible running mates for Trump in 2016 before Trump chose Mike Pence, who was then the governor of Indiana.

Corker appeared to be selecting his words carefully as he spoke to local press at the event in Chattanooga. He joins a handful of GOP senators — including Marco Rubio of Florida, John McCain of Arizona and Lindsey Graham of South Carolina — who have called Trump out by name.

Republican leaders including House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell have put out statements criticizing the conduct the president has exhibited without mentioning his name.

In a statement on Wednesday, McConnell said in part, "There are no good neo-nazis, and those who espouse their views are not supporters of American ideals and freedoms. We all have a responsibility to stand against hate and violence, wherever it raises its evil head."

Corker also said that the president needs to reflect on his role in the country and "move beyond himself, move way beyond himself, and move to a place where daily he's waking up thinking about what's best for our nation."

Trump tweeted support Thursday morning for Kelli Ward, an Arizona doctor and former state senator, who has announced a primary challenge to Sen. Jeff Flake, R-Ariz., a frequent critic of the president.

Flake has strong support from his fellow Republican senators, and Corker said, "The White House would be well-served to embrace the character, the substance of someone like Sen. Flake."

Corker was also asked about a controversy over a Confederate monument in his state. A bust of Nathan Bedford Forrest, an early KKK leader who served as a Confederate general, sits in the Tennessee State Capitol in Nashville. Demonstrators have been pushing state lawmakers and Republican Gov. Bill Haslam to remove it.

"We want to keep our history. We don't want to wash away our history, but let's put it in a museum," Corker said. "And let's have the type of people at public buildings where we go to discuss aspiration things, let's have aspirational figures. Let's have people there who have brought out the best in our nation."

Trump sent out a series of tweets on Thursday decrying efforts to remove Confederate statues and monuments.

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