The US government shutdown has entered its 36th day, making it the longest period of time the American government has been closed.

The shutdown, which began 1 October, was prompted by the inability of Congress to agree to a new funding deal. It has left government workers without paycheque and millions of Americans without essential services.

Democrats and Republicans have remained gridlocked for weeks without any signs of a compromise - though some glimmers of hope are beginning to show.

Majority Leader John Thune, the Senate's top Republican, said his "gut" told him that a possible breakthrough might be on the horizon.

"I just think, based on, sort of, my gut of how these things operate, I think we're getting close to an off-ramp here," Thune said.

The previous government shutdown record happened during Donald Trump's first term as president and lasted 35 days before it ended in 2019.

In the weeks since this shutdown started, the impacts on everyday Americans has worsened.

Thousands of federal workers have already missed paycheques, and there are growing fears of increased impacts to air travel across the US as air traffic controllers and airport staff work without pay.

On Tuesday, Transportation Secretary Sean Duffy told Fox News that some areas of US airspace might have to be closed if the government shutdown continues. About 13,000 air traffic controllers, who are government employees, have been working without pay due to the shutdown.

"If you bring us to a week from today, Democrats, you will see mass chaos," Duffy said. "You will see mass flight delays. You'll see mass cancellations, and you may see us close certain parts of the airspace, because we just cannot manage it because we don't have the air traffic controllers."

The impacts have also been felt by low-income Americans who rely on government services.

One in eight people in the US are dependent on food assistance from the Supplemental Nutrition Assistance Program (Snap) but only a portion of that assistance is being paid out this month due to lapsed funding.

The Trump administration had - at first - said no Snap funds would be dispersed in November but a US court ordered that contingency funds be used to give people some assistance.

Since that Monday ruling, Trump has indicated he may not listen to the judges.

"[Snap benefits] will be given only when the Radical Left Democrats open up government, which they can easily do, and not before," Trump posted on Truth Social. The White House, though, has insisted the administration would follow the court's order.

Members of the US Senate have voted on the same short-term funding bill to reopen the government more than a dozen times with no luck. They tried again on Tuesday to no avail.

That proposal passed the House of Representatives in September.

Democrats have so far refused to support the short-term measure on government funding unless Republicans agreed to extend healthcare subsidies for low-income Americans. Republicans have resisted, accusing Democrats of holding the government hostage over unrelated policy priorities.

If there is a compromise reached, both chambers will have to pass the new legislation, because the initial House-passed bill only keeps the government funded and running until 21 November - a date that is quickly approaching and no longer makes sense, Thune said.

"If we don't start seeing some progress or some evidence of that by at least the middle of this week, it's hard to see how we would finish anything by the end of the week," Thune told reporters on Capitol Hill on Tuesday. "And I think that would be the objective here, is to try and get something that we could send back to the House that would open up the government."

In recent days, there have been indications that some moderate Democrats and Republicans are eager to negotiate and find a deal ahead of Thanksgiving on 27 November.