"The deadly violence we saw overnight in Portland is unacceptable. Shooting in the streets of a great American city is unacceptable. I condemn this violence unequivocally. I condemn violence of every kind by anyone, whether on the left or the right," Xinhua news agency quoted Biden as saying in a statement on Monday.

"And I challenge Donald Trump to do the same. We must not become a country at war with ourselves," the former Vice President said, while accusing his Republican rival of "fanning the flames of hate and division in our society and using the politics of fear to whip up his supporters".

"The job of a President is to lower the temperature. To bring people who disagree with one another together. To make life better for all Americans, not just those who agree with us, support us, or vote for us.

"Donald Trump has been president for almost four years. The temperature in the country is higher, tensions run stronger, divisions run deeper," he added.

The remarks came a day after a motorcade of Trump's supporters drove through downtown Portland, Oregon's largest city, and clashed with people opposing police brutality and racial injustice.

Local police said they are investigating the fatal shooting of one of the pro-Trump protesters.

"It is still early in this investigation, and I ask everyone to give the detectives time to do their important work before drawing conclusions about what took place," Portland Police Chief Chuck Lovell said in a statement.

"This violence is completely unacceptable and we are working diligently to find and apprehend the individual or individuals responsible."

In a series of tweets on Sunday, Trump slammed Portland's Democratic Mayor Ted Wheeler, reiterating his "law and order" message, a central theme of his re-election bid.

"The people of Portland, like all other cities & parts of our great Country, want Law & Order.

"Ted Wheeler, the wacky Radical Left Do Nothing Democrat Mayor of Portland, who has watched great death and destruction of his City during his tenure, thinks this lawless situation should go on forever."

Wheeler, in a press conference on Sunday afternoon, criticized Trump's stance on racial justice, saying that "it's you who has not found a way to say the names of black people killed by police officers".

The shooting also came amid demonstrations in Kenosha, Wisconsin in the wake of the August 23 police shooting of 27-year-old African-American man Jacob Blake.

White House spokesman Judd Deere said on Saturday that the President will visit Kenosha "on Tuesday to meet with local law enforcement and survey damage from the recent riots".

Wisconsin Governor Tony Evers, a Democrat, sent a letter to Trump on Sunday, urging him to reconsider his trip to Kenosha.

"I, along with other community leaders who have reached out, are concerned about what your presence will mean for Kenosha and our state," Evers wrote.

"I am concerned your presence will only delay our work to overcome division and move forward together."

On May 25, George Floyd, an unarmed African-American man, died after a white police officer kneeled on the 46-year-old's neck for nearly nine minutes during an arrest in Minneapolis, Minnesota, which sparked nationwide demonstrations against police brutality and systemic racial inequality, as well as social unrest.

Biden has made race relations a pillar of his White House run in response to the movement, while Trump has focused on violent aspects of the demonstrations and doubled down on his support for police officers.