The partial shutdown seemed certain to drag well into next week, meaning 800,000 federal workers nationwide would continue to go unpaid and some government functions would remain impaired.
Pelosi, on Twitter, shot back at Trump.
Clyburn, who is also at the top of House Democratic leadership, had even proposed a DACA deal as early September 2017.
"What is original in the President's proposal is not good".
"800,000 Americans are going without pay", she tweeted.
The New York Democrat said there is only "one way out" of the shutdown.
"It provides humanitarian relief, delivers real border security and immediately reopens our federal government". But Trump is expected to try to pressure Democrats in other areas.
On Saturday, Trump said Pelosi "is under total control of the radical left".
Trump says Senate Majority Leader Mitch McConnell will bring the proposal for a vote in the Senate this week.
Delaware Sen. Chris Coons, a Democrat seen by White House as a possible partner on immigration negotiations, says he won't support it.
She made clear the plan was a nonstarter in the House after details were reported.
Trump had not directly responded to her call to delay the State of the Union speech till after the shutdown ends.
Florida Gov. Ron DeSantis Demands Repeal On State’s Smokable Marijuana Ban
DeSantis added. "I want people to be able to have their suffering relieved and so I don't think this law is up to snuff". He got right to the heart of the issue, saying "Who am I to judge?" whether patients should smoke marijuana.
The speech was the second time during the shutdown that the President addressed the nation about the immigration crisis.
On this occasion, he sought to strike a diplomatic tone, emphasising the need to work across the aisle.
"This is not a 2000-mile (3,218 kilometre) concrete structure from sea to shining sea".
After his announcement on Saturday, some supporters appeared unhappy with his effort to bridge the divide with Democrats.
Trump's Sunday tweets were a pathetic confession of weakness, not strength. But Trump opposes the bills and McConnell has refused to let any of them come to a vote in the Senate, which is still controlled by Trump's fellow Republicans. "I intend to move to this legislation this week".
The White House yesterday escalated the argument with lawmakers, with the Office of Management and Budget announcing that Congress was effectively grounded unless lawmakers could get White House approval for trips or pay for them themselves.
There had been speculation that Trump would declare a national emergency, which would allow him to build a wall without congressional approval.
Speaking from the White House, Trump said he was offering a "commonsense compromise both parties should embrace".
But he said he would extend protection for "dreamers" - who entered the United States illegally when they were young with their parents - for another three years, allowing them access to work permits.
Administration officials said the protections would apply only to those now in the Obama-era program shielding them from deportation, and the temporary protected status would apply to those who now have it and have been in the USA since 2011.
But anti-immigrant voices also attacked Trump's offer as tantamount to amnesty for the undocumented - a toxic concept for many conservatives. Officials said the exemption would apply to about 300,000 people who now live in the USA under the program.
Also last week, the Trump administration said that new estimates show the cost of the government shutdown will be twice as steep as originally forecast.
Rep. Steve King (R-Iowa), who drew bipartisan condemnation over "racist" white supremacy comments on a radio show, appeared to oppose Trump's proposal, tweeting "NO AMNESTY 4 a wall!" while calling a concrete border wall, a design that Trump no longer advocates, "a monument" to the USA sovereignty.