None of the devices exploded. Hours earlier Trump had complained via tweet that "this "bomb" stuff" was taking attention away from the upcoming election and that critics were wrongly blaming him.
Former CIA director John Brennan, who has been critical of Mr Trump, and billionaire philanthropist George Soros have also been targeted.
Democratic congressional leaders sounded utterly unmoved by the president's remarks. A similar-looking package also may have been sent to Joe Biden, sources told ABC News. The package that included the explosive device and an envelope containing white powder was addressed to former Central Intelligence Agency director John Brennan, a frequent Trump critic.
"The president is certainly not responsible for sending suspicious packages to someone no more than Bernie Sanders was responsible for a supporter of his shooting up a Republican baseball field practice previous year", she said, referring to the shooting that badly injured Rep. Steve Scalise and others.
As for the other suspicious packages, nobody was harmed and the explosives never arrived at the Obamas and Clintons' homes; the packages were intercepted before they got there.
On Tuesday, the Secret Service intercepted a package addressed to Clinton at the home she shares with her husband, former president Bill Clinton, north of Manhattan.
A "pair of unattended packages" on Thursday evening prompted the evacuation of some areas of the Time Warner Center Mall, the location of CNN's studios in New York City. Mr Brennan is a frequent guest on CNN.
The authorities are investigating these incidents and any potential link between them. And the NYPD is telling me that it also is similar to other packages that have been discovered this week and, specifically, it contains a device that was similar to what was recovered from CNN's office in NY yesterday.
None of the devices exploded, and no one was injured. They all also had a half-dozen Forever stamps, computer-printed address labels and return addresses bearing the misspelled name of Wasserman Schultz, who chaired the Democratic National Committee during part of the 2016 presidential campaign.
To the contrary, Reuters says no explosive packages were sent to the White House.
What was in the packages?
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Viral images posted on a Facebook group claim to show migrants on trains and buses heading toward the United States, not walking. Many said they were sick or exhausted, especially the children who toddled or were carried along on the march.
The packages were in manila envelopes with bubble-wrap interior, the Federal Bureau of Investigation said, and each package had six American flag Forever stamps on the envelope.
His latest tweet blaming the media for the anger in the country-and doing this only a day after the pipe bomb scare terror-is simply appalling. After seeing news reports he realized it was similar to other packages in the news and called police at about 4 a.m. on Thursday, according to the official.
"These are not hoax devices", FBI Director Chris Wray said.
How have U.S. political figures responded?
An additional clue: Misspellings from his online posts matched mistakes found on the packages, according to an 11-page criminal complaint that included the formal charges of threatening former presidents and transporting explosives across state lines.
"In these times, we have to unify, we have to come together, and send one very clear, strong, unmistakable message that acts or threats of political violence of any kind have no place in the United States of America", Mr Trump said.
But he says the media "has a responsibly to set a civil tone and to stop the endless hostility and constant negative and oftentimes false attacks and stories".
Bryan Paarmann, special agent in charge of the FBI's Counterterrorism Division and Joint Terrorism Task Force in New York City, said at a press conference on Wednesday that the mailings appear to be political in nature.
CNN chief Jeff Zucker criticized the White House for a "total and complete lack of understanding" of the seriousness of its attacks on the media, as his network's NY bureau was evacuated for five hours Wednesday following the discovery of an explosive device sent there.