Trump's tweet appeared to be part of an effort by his administration to blame far-left activists for the violence gripping the nation after the death of George Floyd in police custody in Minneapolis last week.
It is unclear whether the Trump administration is seriously pursuing the designation through formal channels, which would typically require coordination across multiple federal agencies.
"Any such designation would raise significant due process and First Amendment concerns", said the organization's National Security Project Director Hina Shamsi.
Trump accused the group of starting riots at street protests over George Floyd's death. Police said on Sunday he was in a stable condition. But many protestors and journalists have taken to social media to say police are targeting peaceful protestors and arresting reporters as well.
The closely packed crowds and demonstrators not wearing masks sparked fears of a resurgence of COVID-19, which has killed more than 100,000 Americans so far. The state's governor said on Saturday that he was activating the full Minnesota National Guard for the first time since World War Two. That led authorities to impose a curfew and mobilize the National Guard.
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Lightfoot said the guard is being ordered to Chicago amid "absolutely devastating" looting and destruction downtown.
The widespread resort to uniformed National Guards units is rare, and it evoked disturbing memories of the rioting in USA cities in 1967 and 1968 in a turbulent time of protest over racial and economic disparities.
Protesters clashed with police in New York City - where a police van was torched and several officers were injured - as well as in Chicago, Atlanta and Los Angeles, where more than 500 people were arrested Friday night.
On Sunday night, Minnesota's Department of Corrections Commissioner Paul Schnell echoed Walz in saying he believed white supremacists were attending demonstrations in Minneapolis and St. Paul. "The people that are doing this are not Minneapolis residents", Frey said.
The finger-pointing on both sides of the political spectrum is likely to deepen the political divide in the USA, allowing politicians to advance the theory that aligns with their political view and distract from the underlying frustrations that triggered the protests. The department last year decided not to charge the officer with federal civil-rights violations after a five-year investigation that was started under the prior administration; some prosecutors who worked on the case didn't believe there was enough evidence to prosecute the officer successfully.
The phrase became a rallying point for protesters demonstrating against the killings of unarmed black men and women by police.