Democrats on Monday unveiled legislation designed to re-open the federal government, without providing money for Donald Trump’s border wall.

According to an anonymous aide quoted by the Associated Press, the House is preparing to vote on the package on Thursday, when the new Congress will convene with Democrats in the majority in the lower chamber for the first time since 2010. It will include one bill to fund the Department of Homeland Security at current levels through 8 February, with $1.3bn for border security. Trump has demanded $5bn.

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The package will include six other bills to fund the departments of agriculture, interior, housing and urban development and others closed by the partial shutdown. Some bills have already passed the Senate. Those will provide money through the remainder of the fiscal year, to 30 September.

Monday was the 10th day of the partial government shutdown forced by Trump’s demand for a wall. Hundreds of thousands of federal workers face increasing hardship and key government functions are cast into ever-increasing doubt.

And on Monday afternoon it emerged that some of the most iconic national parks in the western US are closing partially because of problems such as overflowing public toilets and vandalism.

Yosemite national park was closing some of its campgrounds and popular areas. Park officials said that with toilets overflowing, visitors had begun relieving themselves on the side of the road.

In Joshua Tree national park, officials are also closing campgrounds after officials reported that, with only skeleton staff supervising visitors, people were driving off-road illegally and otherwise damaging the park.

Staying open but with barely any staff was “a nightmare scenario” for human and wildlife safety and vulnerable historic sites, John Garder, the budget director of the National Parks Conservation Association, said on Monday.

Meanwhile, over the weekend, confusion reigned about whether Trump actually wants a physical wall along the border with Mexico. Three people close to the president suggested on Sunday he does not. On Monday the president returned to tweeting, stridently, that he does.

Trump campaigned on the promise to build a wall and have Mexico pay for it. He and his allies have claimed a new trade deal with Mexico and Canada, being supposedly better for the US than its predecessor, fulfills the second part of the promise. Over the weekend, the Trump camp tried to explain the president’s thinking on the first.

First, in an interview with the Los Angeles Times, the chief of staff, John Kelly, said Trump abandoned the notion of “a solid concrete wall early on in the administration”.

“The president still says ‘wall’,” said Kelly, whose last day on the job is Monday. “Oftentimes frankly he’ll say ‘barrier’ or ‘fencing’, now he’s tended toward steel slats. But we left a solid concrete wall early on in the administration, when we asked people what they needed and where they needed it.”

Senior adviser Kellyanne Conway then told Fox News Sunday: “There may be a wall in some places, there may be steel slats, there may be technological enhancements. But only saying ‘wall or no wall’ is being very disingenuous and turning a complete blind eye to what is a crisis at the border.”

Senator Lindsey Graham completed the job, telling reporters at the White House after lunch with Trump “the wall has become a metaphor for border security”.

Facebook Twitter Pinterest The US Capitol on 31 December, day 10 of the partial government shutdown. Photograph: Win McNamee/Getty Images

So far, so unclear. And on Monday morning, Trump duly announced: “An all concrete Wall was NEVER ABANDONED, as has been reported by the media. Some areas will be all concrete but the experts at Border Patrol prefer a Wall that is see through (thereby making it possible to see what is happening on both sides). Makes sense to me!”

Graham also said he thought a deal with Democrats might be possible, by which congressional leaders Chuck Schumer and Nancy Pelosi would provide Trump’s money in return for action on the legal status of Dreamers, undocumented migrants brought to the US as children, and other migrant groups.

Saying the president was “open-minded”, Graham added: “Democrats have a chance here to work with me and others, including the president, to bring legal status to people who have very uncertain lives.”

But a previous attempt to reach such a deal broke down, thanks to White House demands.

The president remains at the White House, meeting allies, criticised for not reaching out to Democrats who have happily pointed to an 11 December Oval Office meeting in which Trump said he would be proud to force a shutdown.

A proposed deal for $2.5bn, advanced via the vice-president, Mike Pence, and the Alabama Republican senator Richard Shelby, went nowhere. Conway claimed on Sunday that “the president has already compromised” by dropping his request from $25bn, and called on Democrats to return to the table.

“It is with them,” she said.

Play Video 1:05 ‘I am proud to shut down the government’, Trump tells Schumer – video

Democrats were unmoved.

“It’s clear the White House doesn’t know what they want when it comes to border security,” Justin Goodman, Schumer’s spokesman, told reporters. “The president tweets, blaming everyone but himself for a shutdown he called for more than 25 times.”

Polling shows the public backs the Democrats. But any legislation passed by a Democratic House will have to be backed by the Republican Senate – and Trump.