Congress must vote on a budget to fund the federal government by midnight on Sept. 30.
The shutdown would jeopardize the finances of hundreds of thousands of workers in national parks, museums and other federally funded sites, and could also pose significant political risks for Biden as he runs for re-election in 2024.
Biden on Saturday accused a "small group of radical Republicans" of threatening to paralyze the federal administration next week as a number of its institutions were "shut down" because of funding cuts.
Biden, speaking at a congressional dinner, said he had agreed with Republican House Speaker Kevin McCarthy on the level of public spending for the next fiscal year, which begins Oct. 1.
"Today, there is a small group of radical Republicans who don't want to honor the deal, and all Americans may pay the price."
"Funding government is one of Congress's most fundamental responsibilities. It's time for Republicans to start doing the work America elected them to do. Let's do it."
Congress is currently divided, with Democrats dominating the Senate and the House of Representatives under the control of the Republican opposition.
The White House wants to include $24 billion in military and humanitarian aid to Ukraine in the budget. The measure has the support of Democrats and Republicans in the Senate, but some members of the House of Representatives strongly oppose it.
Republican Representative Marjorie Taylor Green, who is close to former President Donald Trump, said in a video posted on X: "I will not vote for a penny to go to war in Ukraine," stressing, "I put America first. I work for the United States of America. I work for the American people."
House Republican Eli Crane echoed that view.
According to Crane in a video posted on social media, "People in all the country are tired of funding others. We continue to spend, spend, spend with money we don't have."
All of this puts McCarthy in a bind.
Republican Mike Turner told ABC News on Sunday: "He's in a very difficult position because the naysayers keep telling Kevin McCarthy. Bipartisan projects are not put forward. "We don't want you to use Democratic votes to try to avoid shutdowns."
A regular congressional budget vote turns into a bipartisan confrontation, with each side exploiting the prospect of a shutdown to extract concessions from the other before reaching a last-minute solution.
But this year, the confrontation has been exacerbated by unprecedented levels of polarization.
In the Senate, the debate is led by two weighty figures, Democratic Majority Leader Chuck Schumer and, on the Republican side, Mitch McConnell, Minority Leader.
Schumer told CNN on Friday that he and McConnell "strongly support helping Ukraine," adding, "I think a bipartisan majority in the Senate agrees with that."
If no deal is reached, lawmakers could resort to a short-term funding measure, known as a continuing resolution, giving lawmakers a period to find common ground.
This comes just four months after the debt ceiling crisis, in which Washington came dangerously close to the possibility of default, which would have had serious consequences for the US economy and beyond.
As part of the deal, Democrats agreed to limit some spending in a bid to approve the budget.
White House spokeswoman Karine Jean-Pierre confirmed the "deal is an agreement" and blamed Republicans for risking what she called an "unnecessary shutdown."
If the government halts operations, low-income families will no longer receive food aid checks, air traffic will be disrupted and national parks may be shut down.
When closed, civil servants deemed "non-essential" are required to stay at home, and will only receive their salaries when the issue is resolved.
The United States has seen four major closures since 1976. The last was in late 2018 and lasted for five weeks. It cost the U.S. economy $3 billion.
Lawmakers have reiterated their opposition to a repeat but have avoided a closure that may seem difficult.
Republican Representative Tony Gonzalez told CBS on Sunday, "I don't want to see a shutdown," but "I have no doubt that the country is heading toward a shutdown, and everybody has to prepare."