Without passage of funding legislation, parts of the government would close on Wednesday, the first day of the US government's 2026 fiscal year.
Republicans control both chambers of Congress, but a temporary measure keeping the government open would have to amass at least 60 votes in the 100-seat Senate, meaning some Democratic votes would be needed.
But so far, Senate Democrats have rejected a short-term bill, demanding that any legislation undo recent Republican cuts to healthcare programs.
The Republican president has summoned congressional leaders from both parties to a White House sit-down on Monday to discuss funding legislation.
Trump told Reuters on Sunday he believes Democrats want to reach an agreement.
"If they don't make a deal, the country closes," Trump said.
"So I get the impression they want to do something."
Earlier in the day, Republican congressional leaders said they are eager to avert a shutdown.
Republican Mike Johnson, speaker of the House of Representatives, called on Democrats to support a stopgap measure that would fund the government through November 21 while allowing appropriators to continue to hammer out spending bills.
"The only thing we are trying to do is buy a little time," Johnson said on CNN's "State of the Union" program on Sunday.
"We need a serious negotiation," Chuck Schumer, the Senate's leading Democrat, told NBC's "Meet the Press" program.
"Now, if the president at this meeting is going to rant and just yell at Democrats and talk about all his alleged grievances and say this, that and the other thing, we won't get anything done."
The federal government is on the brink of its 15th partial shutdown since 1981 because lawmakers have failed to agree on a plan for discretionary funding - the money that is allocated through the annual congressional budgeting process - for the new fiscal year, or about one-quarter of the $US7 trillion ($A11 trillion) US budget.
If Congress does not act, thousands of federal government workers could be furloughed, from NASA to the national parks, and a wide range of services would be disrupted. Federal courts might have to close and grants for small businesses could be delayed.
Johnson called demands by the Democrats "outrageous," saying government workers and people who rely on government services will suffer.
"It's fine to have partisan debates and squabbles, but you don't hold the people hostage for their services," Johnson said.
Republicans hold a 53-47 majority in the Senate and a 219-213 majority in the House.
Johnson and other Republicans made it clear on Sunday they expect Democrats to agree to the stopgap funding measure and that Monday's White House meeting is not intended to act as a springboard for wide-ranging talks.
A Republican congressional leadership source confirmed the plan for Monday's meeting with Trump is to urge Democrats to vote for the funding resolution so that the House and Senate can have time to complete the appropriations process.
Hakeem Jeffries, the top Democrat in the House, said he is hopeful a shutdown can be avoided but stressed that his party's healthcare concerns must be addressed. Democrats are seeking expanded healthcare subsidies for people who buy insurance through the Affordable Care Act, often called Obamacare.
