The comment came during a brief question and answer session with the President at the White House after Pennsylvania Attorney General Josh Shapiro noted hate crime incidents across the country -- including bomb threats against three Jewish schools and the desecration of a Jewish cemetery in his home state in the last few days.
"Can you tell me what is being done so that the federal and state governments work together to combat this?" Shapiro told CNN he said to Trump.
The President said three things in response, Shapiro recalled.
First, the President condemned the threats and vandalism as "reprehensible." He also noted that he was going to address the matter in his speech to a joint session of Congress Tuesday evening.
But then the President said something that confused Shapiro: "You have to be careful, because the reverse can be true," he recalled the President saying.
Shapiro said Trump suggested this would be done "to make others look bad." Shapiro said the President used the term "reverse" two or three more times.
Other attorneys general later asked Shapiro what Trump meant by that, and Shapiro didn't know what to tell them, he said. The attorneys general did not know what to think.
"I found that statement to be very curious," Shapiro said.
The comment is all the more curious given that law enforcement officials believe many of the threatening calls coming into Jewish Community Centers and Jewish Day Schools have originated overseas.
The White House did not respond to a request for comment. But Jonathan Greenblatt, the CEO of the Anti-Defamation League, said it would be "astonishing if the report is true that the president would posit such a claim. We fully hope that the authorities will continue their ongoing investigation of the matter and have full confidence that the FBI and law enforcement will bring the culprits to justice."
Earlier Tuesday, top Trump adviser Anthony Scaramucci tweeted that "It's not yet clear who the #JCC offenders are. Don't forget @TheDemocrats effort to incite violence at Trump rallies."
Scaramucci linked to a Breitbart story about a James O'Keefe undercover report in which a video suggested that a Democratic activist was sending protesters into Trump campaign rallies where they would film themselves getting beaten up by Trump supporters.
When asked if he was suggesting Democrats were behind the threats to the Jewish community centers, Scaramucci tweeted, "No. I'm saying until we know for sure it's highly irresponsible to jump to conclusions."
Scaramucci said Wednesday "we actually don't know who is behind" the attacks.
"What you're finding is there's a lot of allegations are being made and people are suggesting potentially that it could be Trump supporters behind it or people that are affiliated with the President or his administration," he told CNN's Alisyn Camerota on "New Day." "I think that's categorically very unfair."
Scaramucci did not identify anyone who was making those suggestions and did not immediately respond to a follow-up request for clarification Wednesday morning.
"We have to leave an open question of the source of where these things are coming from," Scaramucci added.
In the last several weeks, there have been at least 100 bomb threats targeting Jewish Community Centers and schools in 33 states in the United States and two provinces in Canada.
UPDATE: This characterization of James O'Keefe's video has been changed in this story.