By Jeremy Diamond, CNN

(CNN) -- President Donald Trump is digging in his heels on his demand for a border wall as the shutdown enters its fourth week, telling aides and allies that he believes he is winning the battle for public support.

Over the last 24 hours, Trump has privately touted a Washington Post-ABC News poll indicating that public support for a border wall has increased to 42% from 34% last year, a source familiar with his comments told CNN.

"He's not going to budge even 1 inch," a source familiar with the President's mindset told CNN.

On Capitol Hill, there are currently no meetings scheduled and no new proposals being traded, aides in both parties say.

But a CNN/SSRS survey published Sunday shows a majority of the public blames the President, with 55% saying he is more responsible for the shutdown than are Democrats in Congress, while 32% say the blame rests mostly with the Democrats.

That poll also shows public opinion is also against Trump on the underlying reason for the shutdown: 56% oppose a wall, 39% favor it, according to the CNN poll.

Still, Trump has cited the Washington Post-ABC News figure as one piece of evidence for why he should allow the shutdown to continue and extend his fight with Democrats, believing they will face more pressure to give him border wall funding as time wears on. He has told aides and allies that he does not plan to budge on his demand that Congress fund a border wall, two sources familiar with his comments said.

In a series of tweets Monday morning, Trump also continued to try and shame Democrats for not negotiating with him over the shutdown, though it's not clear he's offering anything different than when talks crumbled last week.

Trump has been making a point of telling his followers that he's at the White House waiting -- and is clearly frustrated he's not getting credit for remaining in Washington -- as opposed, presumably, to traveling to his Mar-a-Lago resort in Palm Beach, Florida.

The President has repeatedly vowed not to budge from his demand over a border wall, but rejected an idea he floated last week that he would simply bypass Congress and fund a border wall by declaring a national emergency.

"I'm not looking to call a national emergency. This is so simple we shouldn't have to," he told reporters on the South Lawn Monday before leaving for an event in New Orleans.

He also rejected another way out of the stalemate that's resulted in the longest US government shutdown ever -- floated by Sen. Lindsey Graham, R-South Carolina -- that would have temporarily reopened the government while a broader agreement was hashed out with Democrats.