President Donald Trump appears to be running out of options, and patience, after his much-vaunted appeal to end the 30-day US government shut down was roundly rejected by the Democrats.

With polls showing a majority of Americans blame Mr Trump and Republicans for the shutdown impasse, the President said in televised address from the White House Sunday (local time) that he was there “to break the logjam and provide Congress with a path forward to end the government shutdown and solve the crisis on the southern border”.

Mr Trump used the brief speech to launch a new compromise, offering temporary protection from deportations for some undocumented immigrants in exchange for US$5.7 billion ($8 billion) in border wall funding.

That offer was swiftly described by House Speaker Nancy Pelosi as a “non starter” and rejected by opposition Democratic lawmakers, prompting a furious response from Mr Trump.

“It is unlikely that any one of these provisions alone would pass the House, and taken together, they are a non-starter,” Ms Pelosi said in a statement late Sunday.

On Twitter, Mr Trump claimed Ms Pelosi and other Democrats had “turned down my offer yesterday before I even got up to speak”.

He said his opponents “don’t see crime and drugs, they only see 2020 – which they are not going to win. Best economy!”

Nancy Pelosi and some of the Democrats turned down my offer yesterday before I even got up to speak. They don’t see crime & drugs, they only see 2020 – which they are not going to win. Best economy! They should do the right thing for the Country & allow people to go back to work.

— Donald J. Trump (@realDonaldTrump) January 20, 2019

And while his televised overture appeared to offer amnesty for the children of illegal immigrants living in the US, known as “Dreamers”, he backtracked on the apparent offer on social media.

“No, Amnesty is not a part of my offer. It is a 3-year extension of DACA. Amnesty will be used only on a much bigger deal, whether on immigration or something else,” Mr Trump said on Twitter.

“Likewise there will be no big push to remove the 11,000,000 plus people who are here illegally-but be careful Nancy!”

The Dreamers, which refers to undocumented immigrants brought to the US as children, are protected from deportation under the Deferred Action for Childhood Arrivals program.

DACA was put in place under former president Barack Obama.

The Trump administration said in September 2017 it would rescind DACA but it remains in effect under court order.

Mr Trump did not make clear what he was referring to regarding the 11 million people mentioned in his tweet. About 12 million people are living in the US illegally, according to US Department of Homeland Security estimates.

About a quarter of the US government shut down on December 22 over Mr Trump’s demand for $US5.7 billion ($A8.0 billion) to fund a wall along the border with Mexico, which Democrats have refused to consider.

About 800,000 federal workers have been ordered to stay home or work without pay during the shutdown.

Protesters rally outside the NASA Ames Research Center calling for an end to the government shutdown. Photo: Gety

Temporary Protected Status or TPS refers to another class of immigrants: nationals from designated countries affected by armed conflict, natural disaster, or other strife.

Mr Warner, a Virginia Democrat, said reopening the government ahead of border security negotiations was important for preventing future shutdowns.

“If the president can arbitrarily shut down the government now, he will do it time and again,” Mr Warner said.

Mr Warner also said congress should approve pay for federal workers affected by the shutdown before they miss another pay cheque this week.

“Let’s at least pay them on Thursday, so they don’t have to go through more angst,” Mr Warner said.

-with AAP