WASHINGTON — The Postal Service has warned states that it may not be able to meet their deadlines for delivering last-minute mail-in ballots, further fueling the clash over the new postmaster general’s handling of vote-by-mail operations as President Trump continued to rail against the practice.In letters sent in July to all 50 states and the District of Columbia, Thomas J. Marshall, the general counsel for the Postal Service, told most of them that “certain deadlines for requesting and casting mail-in ballots are incongruous with the Postal Service’s delivery standards.”As many states turn to vote-by-mail operations to carry out elections safely amid the coronavirus pandemic, Mr. Marshall urged those with tight schedules to require that residents request ballots at least 15 days before an election — rather than the shorter periods currently allowed under the laws of many states.“This mismatch creates a risk that ballots requested near the deadline under state law will not be returned by mail in time to be counted,” Mr. Marshall wrote.Many states have long allowed voters to request a mail ballot close to the election, but the Postal Service suggested that the large volume of voting by mail at a time of widespread delivery delays meant that states would be better off building more time into their systems.Mr. Marshall said Nevada, New Mexico, Oregon and Rhode Island should not have any trouble, based on their laws, while he requested more information from Vermont and Washington, D.C. The other 45 states, he told them in the letters, face the risk that the timetables set by their laws could leave some voters unable to get their ballots postmarked by Election Day or received by election boards in time to be counted.The letters prompted some states to consider changes that would give voters more time to vote by mail or ensure their ballot would be counted. And their release intensified the criticism directed at the Postal Service and Mr. Trump by Democrats and voting rights advocates, who say the president is deliberately stoking unfounded concerns that voting by mail will lead to fraud and miscounts as a way to cast doubt about the outcome of the election.Word that the letters had been sent across the country, first reported by The Washington Post, came as the Postal Service’s inspector general said she had begun an investigation into the postmaster general, Louis DeJoy, a Republican megadonor and Trump ally who was appointed in May.Democrats in Congress who had urged the inspector general inquiry said they expected it to encompass operational changes to the mail service imposed by Mr. DeJoy and questions about his personal finances, including his ownership of stock in a Postal Service contractor and options in a competitor.In response to the warning letters, some states, including Pennsylvania and Michigan, have called for extensions on counting late-arriving ballots in the November election.“We have asked the Legislature to change Michigan law to allow ballots postmarked by Election Day that arrive within a certain window to be counted,” said Tracy Wimmer, a spokeswoman for Michigan’s secretary of state, Jocelyn Benson.In Pennsylvania, the secretary of state, Kathy Boockvar, asked a court to allow eligible ballots to be counted if they are postmarked by Election Day and received by the following Friday.“The letter made clear that the Postal Service is experiencing significant delays with mail delivery and expects this problem to continue through Election Day,” Ms. Boockvar said in a statement. “The department’s action is simple in its goal — to prevent the disenfranchisement of eligible Pennsylvania voters.”She added that her office continued to have “great confidence” in the vote-by-mail system in the state, which had more than 1.5 million mail-in ballots in the primary, and that the office would continue to advocate for Pennsylvania residents to use voting by mail.Richard Fiesta, the executive director of the Alliance for Retired Americans, which had filed a related lawsuit seeking to ensure voting rights, called the letter “nothing short of outrageous.”“A government agency is taking proactive steps to make voting by mail harder,” he said, adding that the Postal Service was trying to “deprive people the ability to make sure their ballot is counted on time.”In another battleground state, Wisconsin, which suffered through some postal delays during its primary in April, changing the deadlines would require an act of the Legislature. Given that the Republican-controlled body was unwilling to move the April elections amid the peak of the pandemic, there is little hope it would convene a special session now.Instead, the Wisconsin Election Commission is planning to mail voting-information packets along with the absentee ballot request forms to 2.6 million registered voters in September.“It strongly encourages those who choose to vote absentee by mail to request absentee ballots and return them as soon as possible,” said Meagan Wolfe, the administrator of the Wisconsin Elections Commission.After receiving the letter this week, Kim Wyman, the Republican secretary of state in Washington, said she was “highly concerned” and immediately convened a meeting with local and national Postal Service representatives and Washington election lawyers.Ms. Wyman said that after the meeting, it became clearer that most of the letter was a warning about pre-existing deadlines, and that the state would only have to adjust its ballots that go out later than the initial batch. Washington, a vote-by-mail state, mails ballots to its 4.6 million voters about 20 days before the election, but address changes are accepted up to eight days beforehand.“My frustration with both Congress and the White House is this politicization of administrative functions,” Ms. Wyman said. “Attacking the postal process, it undermines people’s confidence in our elections.”In Colorado, a state that votes almost entirely by mail, the secretary of state, Jena Griswold, said she was “very concerned” about the Trump administration’s actions at the Postal Service.“The president has shown a degree of disrespect to this country that is just un-American,” Ms. Griswold said. “There is something that we should all be able to agree on and that is well functioning elections.”Roughly 75 percent of Colorado ballots were returned to a drop box rather than mailed in, and Ms. Griswold said her office would continue to educate voters on their options to return ballots.Ms. Griswold said the local Colorado Postal Service has been a great partner, but she is concerned about the headquarters in Washington and will be investigating.“It is very concerning that there is not clear communication of what is exactly happening from the U.S. Postal Service,” Ms. Griswold said. “I have requested a meeting with the postmaster general. I think he owes Colorado, Coloradans, and every American, an answer of why he is trying to shake it up, fire people, increase costs, in the middle of the pandemic, with the intention to suppress voters.”In California, the state is planning on expanding its already ample and aggressive voter information program.“Am I concerned? Yes. But am I panicking? No,” said Alex Padilla, the California secretary of state. He said that new policies this year that allowed for a 17-day cushion for ballots to be received if they were postmarked by Election Day should help the state weather any major delays at the post office. But voters should still know not to take the potential delays lightly.“Every time anybody attacks the integrity of vote by mail or makes false claims about the integrity of our elections, voter education becomes that much more important and we’re going to be committed to do that,” Mr. Padilla said.Mr. DeJoy has argued that he is modernizing the money-losing agency to make it more efficient. Among his moves have been cuts to overtime for postal workers, restrictions on transportation and the reduction of the quantity and use of mail-processing equipment.In June, union officials received a notice that Postal Service management was removing 671 machines used to sort mail quickly because of a “reduction to letter and flat mail volume.”Mail operations in several battleground states were hit hard by the cuts. On the list for removal were 24 delivery bar code sorters in Ohio, 11 in Detroit, 11 in Florida, nine in Wisconsin, eight in Philadelphia and five in Arizona.In July, Postal Service management sent to employees a “mandatory” order called, “Pivoting for Our Future.” In the memo, the Postal Service said it was banning a practice of postal workers making daily additional trips beyond their initial runs in an effort to save some $200 million.“One aspect of these changes that may be difficult for employees is that — temporarily — we may see mail left behind or mail on the workroom floor or docks,” the memo stated.Last week, the Postal Service put in place a hiring freeze and canceled promotions for nonunionized workers.Mr. Trump has repeatedly made unfounded claims that mail-in voting will lead to widespread fraud, and has pointed to delays in counting in some primary elections this year as evidence that the general election could be chaotic.Even as he assailed the practice, Mr. Trump and his wife, Melania Trump, requested mail-in ballots in Florida, according to Palm Beach County records.Speaker Nancy Pelosi and Senator Chuck Schumer of New York issued a joint statement on Friday condemning Mr. Trump’s “attacks on vote by mail.”“The president’s comments today affirm that no patriotic tradition is immune from his abuse of power,” they said. “The president made plain that he will manipulate the operations of the post office to deny eligible voters the ballot in pursuit of his own re-election.”Luke Broadwater and Hailey Fuchs reported from Washington, and Nick Corasaniti from New York.