His initial comments triggered new warnings from House Speaker Nancy Pelosi and other critics that Trump was using the standoff over the relief bill to further his opposition to mail-in voting -- by hurting the ability of the Postal Service to handle millions of extra ballots in November's election.
Trump has for weeks railed against mail-in voting, making numerous false claims that he repeats regularly in a bid to question the integrity of the upcoming election by asserting that mail-in voting will lead to voter fraud.
Meanwhile, the president on Wednesday requested a mail-in ballot from the elections authority in Palm Beach County, Florida, and the Palm Beach County Supervisor of Elections mailed it to him the same day, according to the office’s website. The state's primary election will be held on Aug. 18.
Trump voting by mail isn't a new phenomenon as it is something he has done before, several times, as well as members of his staff and family.
The new head of the Postal Service, Louis DeJoy -- a Trump donor -- recently made several changes to the agency that could potentially disrupt mail for millions of Americans, particularly absentee and mail-in ballots ahead of Election Day.
"Now, they need that money in order to make the post office work, so it can take all of these millions and millions of ballots," Trump said Thursday on Fox Business Network.
"Now, if we don’t make a deal, that means they don’t get the money. That means they can’t have universal mail-in voting, they just can’t have it," he added.
After his suggestion in the Fox interview fueled allegations from critics that he is seeking to manipulate the postal system for political gain, he was asked later at his daily press briefing whether he was "threatening to veto any legislation that includes funding for the post office."
He responded, "No, not at all."
"What has happened is that's part of a big negotiation, that's a small part of a big negotiation to get more money to people that it wasn't their fault, it was China's fault. And the post office is part of it," he added.
In response to another question about why he doesn't put more resources toward the Postal Service to help it handle mail-in voting as Democrats want, the president said, "If they make a deal, the Postal Service is taken care of, the money they need for the mail in ballots will be taken care of, that doesn't mean we are going to agree to it, but all they have to do is make a deal."
"So, you would sign something that does include funding?" a reporter asked. "Sure, a separate thing, I would do it," he said.
Thursday's remarks follow comments Trump made just the day before, on Wednesday, at his daily pandemic press briefing. Trump said he would not approve $25 billion in emergency funding for the Postal Service, or $3.5 billion in supplemental funding for election resources, citing high costs.
"They don’t have the money to do the universal mail-in voting. So therefore, they can’t do it, I guess," Trump said. "Are they going to do it even if they don’t have the money?"
Democrats have insisted that the proposed $25 billion for the Postal Service came in a plea directly from the agency’s Board of Governors and not from Democrats themselves.
The backlash to the president’s comments was swift.
Pelosi declared Trump as "afraid of the American people" and blamed him for creating obstacles for voters.
"There are people who think that the post office is election central in this election. Maybe the president thinks that, too, and that's why he wants to shut it down," Pelosi said Thursday during her weekly press briefing on Capitol Hill. "The president is afraid of the American people. He's been afraid for a while, he knows that on the legit, it'd be hard for him to win. So he wants to put obstacles of participation."
"…it's a health issue, you shouldn't have to choose between your health and your ability to cast your vote," she added.
Meanwhile, on Thursday, Trump declared: "There's nothing wrong with getting out and voting. You get out and vote. They voted during World War I and World War II."
Presumptive presidential nominee Joe Biden’s campaign also weighed in to cry foul over the president’s statements, calling his accusations an "assault on our democracy."
"The President of the United States is sabotaging a basic service that hundreds of millions of people rely upon, cutting a critical lifeline for rural economies and for delivery of medicines, because he wants to deprive Americans of their fundamental right to vote safely during the most catastrophic public health crisis in over 100 years -- a crisis so devastatingly worsened by his own failed leadership that we are now the hardest hit country in the world by the coronavirus pandemic," Biden spokesman Andrew Bates said in a statement.
"Even Donald Trump's own campaign has endorsed voting by mail and his own administration has conclusively refuted his wild-eyed conspiracy theories about the most secure form of voting. This is an assault on our democracy and economy by a desperate man who's terrified that the American people will force him to confront what he's done everything in his power to escape for months -- responsibility for his own actions," Bates said.
Pelosi, D-Calif., and 174 other Democrats signed a letter sent Wednesday to DeJoy demanding the agency reverses those operational changes they contend would hamper mail-in voting on Nov. 3.
"It is always essential that the Postal Service be able to deliver mail in a timely and effective manner. During the once-in-a-century health and economic crisis of COVID-19, the Postal Service's smooth functioning is a matter of life-or-death, and is critical for protecting lives, livelihoods and the life of our American Democracy," the lawmakers said in their letter.
"The House is seriously concerned that you are implementing policies that accelerate the crisis at the Postal Service, including directing Post Offices to no longer treat all election mail as First Class. If implemented now, as the election approaches, this policy will cause further delays to election mail that will disenfranchise voters and put significant financial pressure on election jurisdictions."
Pelosi and Senate Minority Leader Chuck Schumer also held a meeting with DeJoy on Capitol Hill earlier this month that Schumer later told reporters was "heated."
DeJoy released his own statement this month defending his practices.
"Although there will likely be an unprecedented increase in election mail volume due to the pandemic, the Postal Service has ample capacity to deliver all election mail securely and on-time in accordance with our delivery standards, and we will do so," DeJoy said in a statement.
This report was featured in the Friday, Aug. 14, 2020, episode of “Start Here,” ABC News’ daily news podcast.
"Start Here" offers a straightforward look at the day's top stories in 20 minutes. Listen for free every weekday on Apple Podcasts, Google Podcasts, Spotify, the ABC News app or wherever you get your podcasts.