The US Postal Service (USPS) has warned that millions of mail-in votes may not arrive in time to be counted on the presidential election day, 3 November.
In letters to states across the country last month, the agency said "certain deadlines... are incongruous with the Postal Service's delivery standards".
Critics have blamed the new USPS head - a loyal supporter of President Donald Trump - for a slowdown in deliveries.
A record number of people are expected to vote by mail due to the pandemic.
But on Thursday, Mr Trump said he was blocking additional funding for the USPS to help with election issues, because he opposed mail-in voting.
He has repeatedly said mail-in ballots will lead to voting fraud - and give a boost to his rival Democrat Joe Biden. Experts say the mail-in voting system - which is used by the American military and by Mr Trump himself - is safe from tampering.
Former President Barack Obama strongly criticised what he described as Mr Trump's "attempts to undermine the election", writing on Twitter that the administration was "more concerned with suppressing the vote than suppressing a virus".
Meanwhile, Congress's two top Democrats - Nancy Pelosi and Chuck Schumer - called on the president to stop his "assault" on the postal service and "allow the 2020 election to proceed without his sabotage tactics".
Their comments come as a poll by Axios/ Survey Monkey found that three quarters of Republican voters plan to vote in person, while more than half of Democratic voters plan to use a mail vote.
Private delivery services Fedex and UPS have both rejected calls to help ease the pressure on the postal agency.
Meanwhile, the USPS has reportedly begun removing mail sorting machines - many of which would normally be used to process ballots during the election - according to Vice.
What did the USPS say?
The USPS, which has long been in financial trouble and carries about $160bn (£122bn) in debt, sent letters to states across the US in July. It warned that it could not guarantee that all votes cast by mail would arrive on time to be counted. At least 15 states have received a letter, according to NBC News.
In a letter to Pennsylvania's secretary of state, the USPS said mail-in ballots requested one week before the 3 November election - allowed under the state's election laws - may not reach their destination on time because the state's deadlines are too tight for its "delivery standards".
USPS General Counsel Thomas Marshall said a "mismatch" between Pennsylvania's laws and the mail system's delivery capabilities "creates a risk that ballots requested near the deadline under state law will not be returned by mail in time to be counted under your laws as we understand them".
The letter was made public on Thursday as Secretary of State Kathy Boockvar asked the state's supreme court to allow ballots to be counted as long as they were received up to three days after the election. Currently, votes are discarded if they are received after election day.
Pennsylvania is a battleground state, which Mr Trump won by less than 1% in the 2016 election. Other battleground states, including Florida and Michigan, also received letters, according to US media reports.
The Democratic governor in Pennsylvania's neighbouring New Jersey announced on Friday that the state would pre-emptively send ballots to every registered voter in the state. The process of sending out ballots is known as universal mail-in voting, and has been adopted in nine other US states.
By David Willis, BBC North America correspondent
American voters have been here before of course. In the year 2000, the entire US presidential election result was decided by a few hundred contested votes in the state of Florida, after ballots were scrutinised and sometimes rejected, and the process dragged on for weeks.
President Trump has said he wants a clear result on election night, not a contest that drags on through the courts. But by blocking the allocation of badly needed funding to the beleaguered US postal service, Mr Trump is potentially paving the way to a series of drawn-out legal battles that could stretch on for weeks.
According to reports here, conservative groups are marshalling a massive legal effort aimed not only at limiting the allocating of postal ballots, but challenging results that prove unfavourable to them on election night.
In possibly the harshest criticism of his successor to date, Barack Obama has accused Donald Trump of attempting to "kneecap" the US postal service in order to discourage people from voting, while Mr Trump's Democratic rival Joe Biden has accused the man he is seeking to replace of launching an "assault on democracy".
What's the background?
Critics say changes made by the new Postmaster General Louis DeJoy - a major Republican donor - like clamping down on overtime and halting late delivery trips have led to an increase in mail waiting times.
But Mr Trump told Fox News he was blocking additional funding for the financially troubled agency, because he opposes mail-in voting.
"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," he said. "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."
Amid the funding controversy, the 300,000-member National Association of Letter Carriers union on Friday endorsed Mr Biden, warning that the "very survival" of the USPS was at stake.
Mr Trump's campaign has not yet responded.