Republican Senator Lindsey Graham has turned on Donald Trump's administration, blaming senior White House adviser Stephen Miller for the lack of progress towards ending the ongoing government shutdown.

Mr Graham said late on Sunday that he expected a "breakthrough" on the shutdown vote, but that "as long as Stephen Miller is in charge of negotiating on immigration, we're going nowhere."

"He's been an outlier for years," he said of Mr Miller.

The senior Trump adviser was once a staffer to current Attorney General Jeff Sessions and is known for his anti-immigration and staunch conservative views.

"Every time we have a proposal it is only yanked back by staff members," Mr Graham lamented, adding that though he does "enjoy working with them", White House staff like Mr Miller are "making it very difficult."

The President's wide-ranging immigration plan, authored by Mr Miller, made funding for both the President's proposed US-Mexico border wall and a crackdown on US cities unwilling to assist federal authorities in rounding up undocumented immigrants - known as sanctuary cities - non-negotiable parts of the deal.

Mr Graham's attack on Mr Miller comes just weeks after he defended the President from media criticism, telling CNN in December: "What concerns me about the American press is this endless, endless attempt to label the guy as some kind of kook not fit to be president.”

The senator had refused to support Mr Trump's presidential campaign in 2016, labelling the candidate a "kook" who was "unfit for office".

He later heavily criticised the President's response to the racially motivated protests in Charlottesville last summer, accusing him of stoking tensions.

As the shutdown extends into the working week, thousands of f ederal employees were being told on Monday to stay at home or work without pay.

Senators negotiated late into Sunday night in a bid to reach an agreement that would have reopened federal agencies before Monday morning, but failed to reach a deal.

Immigration and border security funding continue to be the primary sticking points preventing progress towards ending the shutdown.

The key policies at stake are Mr Trump's proposed border wall with Mexico and the Deferred Action on Childhood Arrivals (DACA) programme, which allows people brought to the US as undocumented immigrants when they were minors to remain in the US legally and avoid deportation.

Democratic Senator and minority senate leader Chuck Schumer blamed Mr Trump for the shutdown, claiming the President had walked away from “not one, but two bipartisan deals.”

“If he had been willing to accept any one of these deals we wouldn’t be where we are today,” he said.

“Americans know why the dysfunction is occurring: a dysfunctional president. Hence, we are in a Trump shutdown.”

He had harsh words for other Republicans too, whom he accused of causing “chaos” and “gridlock”.

Some Republicans have forgotten “the lessons of the founding fathers and shown they don't know how to compromise," Mr Schumer said.

"Not only do they not consult us, they can't even get on the same page as their president - the president from their own party."

Mr Schumer, one of the most powerful Democrats in Washington, has been one of the most vocal opponents of the President in the last three days after a meeting between the two men on Friday failed to end the deadlock.

White House Press Secretary Sarah Huckabee Sanders said the senator’s memory of the meeting – in which Mr Schumer claims Mr Trump showed a willingness to compromise but later failed to press his party to accept it – is “hazy” and his recollection “false”.

Some Republicans have pinned the blame for the shutdown on Mr Schumer, spreading the hashtag “Schumer Shutdown” on social media.

He is accused of being captive to liberals and advocacy groups who are pushing for a solution for immigrants currently protected under the DACA programme.

There are approximately 800,000 people - so-called 'Dreamers' - who have benefited from DACA in the US since its beginning in 2012 under the Obama administration. Potential participants had to have been brought into the US before 15 June 2007 and be younger than 16 when they arrived.

Many Democrats have said rescinding protection from Dreamers is not only against the principal of a diverse and welcoming country but also does not make economic sense, especially with Mr Trump's 'America First' policy on American jobs and manufacturing.

A vote on re-opening the government will be held on Monday at noon local time (5pm UK time). If it passes, the government will extend funding until 8 February, allowing immigration bills to be brought to the floor of the Senate to be voted upon before 5 March, when DACA participants lose their legal status.