WASHINGTON — President Trump stirred new questions on Thursday about whether he would seek to hold up new money to the Postal Service to impede mail-in voting this fall in the middle of the pandemic.

Repeating the unfounded claim that the election could be riddled with fraud if mail ballots were widely used, he made clear that he opposed Democratic demands for additional funding for both the post office and election security measures because of his opposition to mail-in voting. Still, he left open the possibility that he could come to a deal as part of a larger negotiation over a new round of economic stimulus.

“We have to have an honest election,” the president said when pressed on mail voting at a news conference in the White House Briefing Room. “And if it’s not going to be an honest election, I guess people have to sit down and think really long and hard about it.”

His statements added to growing alarm among Democrats and voting rights proponents that Mr. Trump is intent on undercutting mail balloting and sowing discord and confusion over the result of the election. They came amid growing scrutiny of the postmaster general, a Republican megadonor.

Mr. Trump stressed the issue across the day in a series of inconsistent remarks in which he veered between a hard line against mail-in balloting and signals of some openness to an accommodation.

“They need that money in order to have the post office work so it can take all of these millions and millions of ballots,” Mr. Trump said Thursday morning in an interview on Fox Business, referring to Democratic demands. “If they don’t get those two items, that means you can’t have universal mail-in voting because they’re not equipped to have it.”

The dispute over the money for the post office and election security is part of the larger partisan standoff over another round of economic stimulus in response to the coronavirus pandemic. Talks between the two sides collapsed at the end of last week, and although the administration made an overture on Wednesday, no serious new negotiations are underway. Both chambers have now left Washington until September, after the Senate formally adjourned on Thursday.

Mr. Trump suggested that without agreement on the larger package, Democrats would have to give up on widespread use of mail balloting. He regularly states that mail-in voting leads to election fraud.

“If we don’t make a deal, that means they don’t get the money,” Mr. Trump said. “That means they can’t have universal mail-in voting, they just can’t have it.”

But when asked later at his news conference if he would veto any legislation that contained more money for the post office and election security, Mr. Trump said no, suggesting that he was using the issue in part as a negotiating chip with congressional Democrats on the larger stimulus package.

“All they have to do is make a deal,” he said. “If they make a deal, the Postal Service is taken care of, the money they need for the mail-in ballots would be taken care of — if we agree to it. That doesn’t mean we are going to agree to it, but all they have to do is make a deal.”

The Postal Service has said regularly that it can handle any surge in volume from mail balloting this fall. More than three-quarters of voters will be able to cast their ballots by mail this year.

Mr. Trump has assailed the money-losing Postal Service in recent months, at the same time warning that voting by mail will lead to fraud, lost or stolen ballots and long delays in determining a winner. (On Wednesday, both the president and the first lady, Melania Trump, were mailed ballots, Palm Beach County, Fla., election records show. It was first reported by The Palm Beach Post. Late Thursday, Judd Deere, a White House spokesman, said, “The president supports absentee voting, not universal mail-in voting, which contain several safeguards that prevent fraud and abuse.”)

The appointment as postmaster general in May of Louis DeJoy, a Trump campaign contributor with significant financial interests in the Postal Service’s competitors and contractors, has prompted further concerns about the politicization of the agency, particularly after Mr. DeJoy put in place policy changes that have slowed mail delivery in some areas.

Mr. DeJoy has kept tens of millions of dollars invested in XPO Logistics, a Postal Service contractor for which he was a board member, first reported by CNN on Wednesday. However, he sold his stake in United Parcel Service, a major rival for the post office, in June, according to financial disclosures.

Shortly after he divested between $100,000 and $250,000 in Amazon stock the same month, he bought $50,000 to $100,000 in stock options for the company. Amazon, a frequent subject of Mr. Trump’s attacks, is a major competitor to the Postal Service in package delivery.

Following the revelations, Senator Elizabeth Warren, Democrat of Massachusetts, called for an investigation into Mr. DeJoy’s holdings.

A spokesman for the Postal Service said that the postmaster general was not required to divest his assets unless a conflict of interest arose.

“I take my ethical obligations seriously, and I have done what is necessary to ensure that I am and will remain in compliance with those obligations,” Mr. DeJoy said in a statement.

The president’s remarks on Thursday connected his objections to additional aid for the agency and his aversion to mail-in voting more directly than he has previously. Critics of the administration said it was further evidence that Mr. Trump seeks to delegitimize the November vote before it happens.

“The president is afraid of the American people,” Speaker Nancy Pelosi of California said at her weekly news conference. “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.”

She added: “But we do not agonize. We organize.”

A spokesman for Mr. Trump’s presumptive Democratic opponent, former Vice President Joseph R. Biden Jr., called it a cynical attempt at disenfranchisement.

“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,” said Andrew Bates, a spokesman for the Biden campaign.

Officials have repeatedly said that the agency is equipped to handle a surge in mail-in voting, with Mr. DeJoy telling a board meeting last Friday that the Postal Service “has ample capacity to deliver all election mail securely and on time in accordance with our delivery standards.”

But voting activists said that Mr. Trump’s remarks simply made clear what they already suspected: that the president was attacking the post office to undermine the election. Tammy Patrick, an expert on mail-in voting and senior adviser at the Democracy Fund, a nonpartisan grant-making foundation, maintained that funding was not intended to implement a “universal vote by mail,” as the president put it, but rather a secure option for voters amid the pandemic.

Wendy Weiser, the director of the Democracy Program at the Brennan Center for Justice, a New York-based research organization, said Mr. Trump’s comments effectively throw “the ball into Congress’s court” to provide the necessary money. Any funding bill, however, would require Mr. Trump’s signature to become law.

Negotiations over another coronavirus relief package have sputtered to a halt. While Republicans have insisted on a $1 trillion measure, Democrats have pushed to infuse at least $2 trillion into the American economy and include money for state and local governments, food assistance programs and for election security and the Postal Service.

In addition to new funding for the Postal Service, Ms. Pelosi and Senator Chuck Schumer of New York, the Democratic leader, have called for legislative language that would counter some of the operational changes Mr. DeJoy has instituted.

Democrats originally called for $25 billion for the Postal Service over three years, a figure Ms. Pelosi has said they reached after discussions with the bipartisan board of governors.

But Democrats have since raised the prospect of lowering the amount to $10 billion over one year, according to a person familiar with the discussions, who asked for anonymity to disclose details of the private talks.

At least one Republican has also expressed support for providing some additional money to the agency.

“I do disagree with the president on the need to support the Postal Service,” said Senator Susan Collins of Maine, one of a number of vulnerable Republicans up for re-election in November.

“There’s no doubt that the Postal Service has long-term financial challenges, and those need to be dealt with,” she added. “But now is not the time to be cutting back services.”

Katie Glueck, Michael D. Shear and Annie Karni contributed reporting.