With time dwindling before a new Congress is sworn in, Donald Trump refused to budge in his demand for billions for a border wall, narrowing prospects for a swift end to an increasingly damaging government shutdown.

His claim that Democrats are to blame for the deaths of migrant children in US custody, meanwhile, sparked a nasty political controversy.

Democrats will take control of the House of Representatives on 3 January. Through Sunday, Trump stuck to his strategy of trying to foist blame for the shutdown, in its ninth day, on the opposition party.

Senior White House adviser Kellyanne Conway said the president was waiting for Democrats to negotiate, telling Fox News Sunday: “It is with them.” Amid criticism that Trump has not spoken to the incoming speaker, Nancy Pelosi, since 11 December, Conway told CNN’s State of the Union the president was “in the White House. He’s in Washington ready to negotiate.”

Democrats have accused Trump of derailing government to advance a pet policy. On Twitter on Sunday Trump fired back, saluting “great work by my administration over the holidays to save coast guard pay … no thanks to the Democrats who left town and are not concerned about the safety and security of Americans!”

He later seemed to attempt a joke, saying “a ten foot wall” around Barack Obama’s Washington home showed “the US needs the same thing, slightly larger version!”

On Saturday, Trump noted that his plan had failed to gain the support it needs in the Senate: at least 10 Democrats.

“Now we have to do it the hard way, with a Shutdown,” he wrote, seemingly adding fuel to Democratic claims he is to blame. In that last meeting with Pelosi, and Senate minority leader Chuck Schumer, the president said he would be proud to force a shutdown. Last week, Democrats said they had rejected a White House offer, made through the vice-president, Mike Pence, to accept $2.5bn, down from $5bn. It is not clear if Trump would have supported that deal.

Richard Shelby of Alabama, the Republican chair of the Senate appropriations committee, told CBS’s Face the Nation: “The president made some proposals through the vice-president. I made some proposals to Senator Schumer … but right now we’re at a standoff, and I think that’s not good for the Senate, the House or America.”

Later, Senator Lindsey Graham of South Carolina, a Trump ally, emerged from the White House saying he was optimistic a deal could be done – if Democrats agreed to provide money for barriers at the border in return for action to protect some undocumented migrants.

“The president was upbeat, he was in a very good mood, and I think he’s receptive to making a deal,” Graham said, adding that Trump found the idea about undocumented migrants “interesting”.

“I don’t see Democrats giving us more money unless they get something,” Graham said, adding that he had “never been more encouraged that if we can get people talking we can find our way out of this mess and that would include around $5bn for border security, slash wall, slash fencing whatever you want to call it in areas that make sense.”

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Graham was echoing Republican talking points, that Trump’s demand for a “wall” does not in fact constitute a demand for a continuous barrier. Outgoing White House chief of staff John Kelly admitted in an interview published on Sunday the proposal for increased border security was “not a wall”.

But Democrats, when not pointing to Trump’s campaign promise that Mexico would pay for the wall, contend there is little need for any form of barrier. Voters appear to agree: at least 56% of Americans oppose Trump’s idea, according to a Harvard CAPS/Harris poll released on Friday.

Furthermore, Democrats have little reason to concede anything. They won overwhelming victories in the November elections by opposing Trump’s immigration proposals. A Reuters/Ipsos poll this week found that 47% of Americans blamed Trump for the shutdown and 33% blamed Democrats.

Controversially, Trump tweeted on Saturday that “any deaths of children or others at the border are strictly the fault of the Democrats and their pathetic immigration policies”.

Two Guatemalan children have died in US custody. The causes of their deaths remain disputed but amid criticism of the president, Conway sought to turn the tables, telling CNN: “I don’t like some of these Democrats using these deaths as political pawns.”

She added: “The president does not want these children taking these perilous journeys. Some of them are paying the ultimate price.”

Former Virginia governor Terry McAuliffe, a Democrat openly considering a White House run, told CNN the president’s use of the deaths was “the lowest act I have ever seen [from] any president in the history of this country”.

On ABC’s This Week, commissioner of US Customs and Border Protection, Kevin McAleenan, said the deaths were “absolutely devastating”.

“Our agents did everything they could, as soon as these children manifested symptoms of illness, to save their lives,” he said.

Asked if the federal government bore any responsibility, he said it was “a multi-faceted problem that requires a multi-faceted solution”. He also called the situation at the border a “humanitarian crisis” and said children were “coming through a system built for adults who are violators of the law”.

“What we’ve done immediately,” he said, “is that we do medical checks of children, 17 and under, as they come into our process. That’s not a capacity we’ve had.”

McAleenan spoke positively of aid to central American countries, with whom he said the US should work. In a Friday tweet, Trump threatened to cut off aid to Honduras, Guatemala and El Salvador.

While the shutdown goes on, hundreds of thousands of federal workers remain without pay and key functions at agencies covering the environment, agriculture and other sectors are affected. This week, Trump claimed without evidence that most workers affected “are Democrats”.

Contributors

Tom McCarthy and Martin Pengelly

The GuardianTramp

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