Trump's remarks came after two "potential explosive devices" sent to the homes of Obama and Clinton were intercepted and proactively detonated on Wednesday in what New York police called an attempted terrorist attack.
Explosives were also found in package sent to the US news network CNN, after which Time Warner building in New York City, where the CNN bureau is located, was evacuated, the US Secret Service said.
"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," Trump said at a White House event on opioid crisis.
Earlier, he was briefed by officials from the FBI, Department of Justice, Department of Homeland Security, and the US Secret Service.
"This egregious conduct is abhorrent to everything we hold dear and sacred as Americans," Trump said.
He said his administration will spare "no resources or expense in this effort" to investigate who sent the packages.
"A major federal investigation is underway. This egregious conduct is abhorrent to everything we hold dear and sacred. We're extremely angry, upset, unhappy about what we witnessed this morning and we will get to the bottom of it," he said.
In his brief remarks before signing a bill on opioid crisis in the East Room of the White House, Trump, without directly naming top democrats - Obama and Clinton - and CNN, referred to the explosive devices as "suspicious packages".
The President, however, parried a question on whether the incident is being considered an attempted terrorist attack, as described by New York police.
"The safety of the American people is my highest and absolute priority," Trump said.
"The full weight of our government is being deployed to conduct this investigation and bring those responsible for these despicable acts to justice," he said.
The explosives being described by the New York Police Department as pipe bomb, has been received by several other important personnel, including at least two Democratic lawmakers.
The development comes two weeks before the crucial mid-term polls on November 6.
First Lady Melania Trump, speaking at the same event, said the attempted attacks on Clinton, Obama, their families, public officials, individuals and organisations, cannot be tolerated.
"We can not tolerate those cowardly attacks and I strongly condemn all who choose violence," she said.
"I'm grateful to the Secret Service as well as to the local and federal law enforcement for all they do on a daily basis to keep us safe and encourage people across the country to choose kindness over hatred," she added.
Responding to the development, Hillary Clinton said it's a sign of a "troubling time" in America.
"It is a troubling time, isn't it? It's a time of deep divisions and we have to do everything we can to bring our country together. We also have to elect candidates who will try to do the same," she said at a campaign event in Florida.
Clinton, who was the Democratic presidential candidate in 2016, thanked the Secret Service for intercepting the package.
"We are fine, thanks to the men and women of the Secret Service who intercepted the package sent to us long before it made its way to our home," she said.
The comments were Trump's most extensive since the threats were reported.
He previously offered a tweet saying he "wholeheartedly" agreed with Vice President Pence's condemnation of the threats.
US Capitol Police are investigating a suspicious package addressed to Democratic Congressman Maxine Waters, multiple news outlets said.
Trump has singled out each individual for criticism since taking office, and he regularly derides CNN as "fake news."
CNN said the package sent to it was addressed to former CIA director John Brennan.
It said that the package has been removed from its New York building and another security sweep is going on.
The FBI said the package intended for the former first lady and secretary of state Clinton was found in the vicinity of her residence in Chappaqua, New York, but offered no additional comment, citing the ongoing investigation.
The Clintons have lived in Chappaqua, about 48 km north of New York City, since former president Clinton left office in 2001.