Politicians in the United States have discussed a temporary funding proposal aimed at ending a record-long partial federal government shutdown, after two competing measures to terminate the impasse fell short in the Senate.

A bill backed by Republican President Donald Trump to end the shutdown by funding a US-Mexico border wall, and a separate measure supported by Democrats to reopen shuttered agencies without such funding did not get the votes on Thursday required to advance in the 100-member chamber.

The shutdown, now in its 34th day, was triggered last month by Trump by demanding $5.7bn for the wall as part of any legislation to fund about a quarter of the government.

It has left 800,000 federal workers as well as private contractors without pay and struggling to make ends meet, with the effects on government services and the economy reverberating nationwide.

Shortly after Thursday's votes, a possible way out of the deepening crisis began to take shape when Senate leaders huddled in private to discuss the proposal to fund lapsed federal agencies for three weeks to allow for negotiations over border security.

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Asked if he would support the plan, discussed in private by Senate Majority Leader Mitch McConnell and top Senate Democrat Chuck Schumer, Trump was noncommittal, saying he would still want to fund his wall.

"If they come to a reasonable agreement, I would support it," he said, but added: "We have to have a wall in this country."

Sarah Sanders, a White House spokeswoman, also warned in a statement that a three-week funding bill would "only work if there is a large down payment on the wall".

But Lindsey Graham, a Republican senator who often has the president's ear, said he discussed with Trump the proposal for a stopgap funding bill.

"All of us believe that if we had three weeks with the government open, with all the discord coming from a shutdown, that we could find a way forward to produce a bill that he would sign," Graham said on the Senate floor.

He added in a Twitter post that Democratic priorities for disaster relief were also included, "showing good faith from both sides".

Political acrimony

Trump had backed the Senate measure which would have reopened the government, funded the wall and included some immigration policy changes. It earned 50 votes to 47 against, but it needed 60 to advance.

The competing proposal by Senate Democrats that would open government through February 8 without funding Trump's wall, and leave room for border security negotiations, also failed to move forward by a similar tally.

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House of Representatives Speaker Nancy Pelosi, a Democrat, told reporters earlier she was willing to meet Trump to discuss the shutdown.

Her comments came one day after she essentially withdrew an invitation for Trump to give his State of the Union in the House chamber next Tuesday, saying that would not happen until the shutdown ended.

Trump, who had planned to come despite the shutdown and considered giving the speech at another venue, conceded late on Wednesday and said he would deliver the speech in the House in the "near future".

Though politically bruised, Trump stood firm on his demand for billions of dollars in funding for the border wall as part of an overall budget package.

"We will not Cave!" he tweeted at Pelosi before the Senate votes.

Dire warnings

Trump has touted Republican unity during the longest-ever halt to federal operations in the US. But in a sign of cracks in that resolve or a desire for compromise, six Republican senators voted with Democrats on their measure to temporarily reopen government agencies without money for Trump's wall.

They included freshman Senator Mitt Romney, his party's 2012 presidential nominee.

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"Democrats have said they're not willing to negotiate unless the government's open. Well they tried their effort. I voted for it. It didn't happen. Now they've got to negotiate," Romney said.

A Reuters/Ipsos poll last week found more than half of Americans blamed Trump for the shutdown even as he has sought to shift blame to Democrats after saying last month he would be "proud" to close the government for border security.

The air travel industry issued dire warnings, including one from the National Air Traffic Controllers Association (NATCA).

"In our risk-averse industry, we cannot even calculate the level of risk currently at play, nor predict the point at which the entire system will break," NATCA President Paul Rinaldi said in a statement.

"It is unprecedented."

Most Democrats oppose direct funds for wall construction, but some have begun publicly advocating for negotiating a solution that includes boosted border funding, including for border structures.

And Pelosi herself said on Thursday that "we will have some of our proposals for what comes next" in upcoming legislation offerings.