Mayor Ted Wheeler on Monday called on the federal government to revoke the permit for a rally set for Sunday, and to deny permission for a second rally on June 10.
The two gatherings -- backed by different organizers -- are planned for a park on federal land, which gives federal authorities control over issuing permits.
"Our city is in mourning, our community's anger is real, and the timing and subject of these events can only exacerbate an already difficult situation," Wheeler posted on Facebook.
Two men suffered fatal stab wounds, and a third man was seriously injured, when they intervened in an incident on Friday on a Portland commuter train. Police said the men were trying to stop a man who was yelling anti-Musllim slurs at two young women -- one of whom was wearing a hijab -- on the train.
The FBI is assisting Portland police, who are leading the investigation and are trying to determine whether Jeremy Joseph Christian, charged in the killings, could also be charged with federal hate crimes. The attacks occurred hours before the start of Ramadan, a month-long Muslim holy period of fasting, prayer and charity.
Wheeler said the city would not issue permits for the June 4 and June 10 rallies, according to a statement. But the federal government has already granted a permit for the June 4 rally, the mayor said. He urged the groups to cancel the rallies and asked their supporters to stay away.
"My main concern is that they are coming to peddle a message of hatred and of bigotry," Wheeler told reporters, referring to organizers of the two rallies. "They have a First Amendment right to speak, but my pushback on that is that hate speech is not protected by the First Amendment to the United States Constitution."
Organizer: Suspect 'has nothing to do with us'
CNN affiliate KATU
said the "Trump Free Speech Rally Portland," is scheduled for Sunday.
Joey Gibson is organizer of that rally. His Facebook page says the effort is "about fighting corruption and big government with the strength and power of love."
Gibson told CNN Monday that the mayor "is using this as an opportunity to use these two dead people to silence us." Gibson also distanced his group from the train-killings suspect, Christian.
Gibson said Christian showed up an April 29 free speech march he organized in Portland "with a bat yelling and screaming, cussing at people, using derogatory names."
"Jeremy Christian has nothing to do with us. He hated us, he threatened me. We did everything we could to kick him out. We didn't want him with us," Gibson said.
In a video of the April free speech rally, Christian is seen shouting at people. At one point, Christian blurts a racial epithet, according to the video from reporter Doug Brown of The Portland Mercury
weekly newspaper show.
Chrisitan, 35, identifies himself in one of the videos, and he's also seen carrying the baseball bat, which he handed over to officers who approached him.
Christian has been charged with two counts of aggravated murder and one count of attempted murder, all felonies, in Friday's attack aboard a Portland commuter train. The aggravated murder charges have the death penalty as a possible sentence. He faces several misdemeanors including two counts of second-degree intimidation.
Christian is scheduled to be arraigned on Tuesday, police said.
Police said Ricky John Best, 53, of Happy Valley, and Taliesin Myrddin Namkai-Meche, 23, were killed in the attack.
The third man, Micah David-Cole Fletcher, 21, suffered serious but non-life-threatening wounds, police said.
Earlier, in an interview with CNN, Wheeler called the three men heroes.
"Their actions were brave; they were selfless, and it could serve as an example and inspiration to all of us," Wheeler said.
When asked if he identified with Wheeler's "alt-right" description, Gibson said: "When someone can explain to me what alt-right means, I'll answer it. I'm a libertarian. I'm not right wing, I'm for gay marriage, I'm against the drug wars."
The so-called "alt-right,"
is a far-right movement that has been linked to white nationalism, racism, misogyny and anti-Semitism.
ACLU: Government cannot shut down free speech
Gibson said he will not cancel the June 4 rally.
"If I cancel this rally and we don't have a permit, you're talking about hundreds of people just showing up in the park with no leadership, no voice of reason," he said in a Facebook post.
The June 10 protest is called #MarchAgainstSharia, KATU reported.
The ACLU of Oregon, in a tweet on Monday, said: "The government cannot revoke or deny a permit based on the viewpoint of the demonstrators. Period."
"We are all free to reject and protest ideas we don't agree with. That is a core, fundamental freedom of the United States," the ACLU said.