Key Democrats on Saturday expressed openness to President Donald Trump's proposed wall on the U.S. border with Mexico, a dramatic turnaround for the party as it seeks to extend protections for people who were illegally brought to the U.S. as children.

Senate Minority Leader Chuck Schumer said early Saturday morning that, in a Friday meeting with the president, he offered to put the wall "on the table" in a potential deal to avoid a government shutdown.

Later Saturday, Illinois Democratic Rep. Luis Gutierrez, a longtime opponent of Trump's wall, told reporters that he would back off his opposition to the president's plan for the barrier, in order to protect recipients of the Deferred Action for Childhood Arrivals program, or DACA.

"It's not about a wall. We'll build him a wall. Tell us how high you want it. But free the Dreamers," the lawmaker said, according to journalists on Twitter.

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Earlier this week, Gutierrez told the New York Times that Trump's chief of staff, John Kelly, told the House Hispanic Caucus that "a 50-foot wall from sea to shining sea isn't what we're going to build." Trump later insisted that he had not changed his mind about the wall.

During the 2016 election, a wall along the American border with Mexico was a cornerstone of Trump's populist, nationalist pitch to voters, and the idea remains a point of contention in the president's relationship with Congress.

Yet Democrats have largely resisted Trump's calls for billions of dollars to build the barrier, and any indication they are willing to budge is likely to anger their liberal base.

Schumer revealed his offer to Trump in remarks on the floor of the Senate minutes after the chamber failed to pass legislation that would have averted a shutdown.

On Friday, Schumer had a 90-minute meeting with Trump in the Oval Office. Only Kelly and Schumer's chief of staff, Michael Lynch, were in the room with the senator and the president. Afterward, Schumer had a positive take on the meeting, although he did not say a deal was struck.

Still, the New York Democrat said Trump had walked away from the proposition. The White House has said that it would prefer addressing immigration issues separately from a measure to fund the government.

Saturday afternoon, White House budget chief Mick Mulvaney, in an impromptu press briefing, said Schumer's offer to fund the wall fell well short of the money Trump has been seeking.

Mulvaney, citing Kelly, said that Schumer told Trump he would agree to "all the money for the wall," which Trump took to mean the full $20 billion that building a partial border wall is expected to cost. But what Schumer was actually talking about was the $1.6 billion the White House had already sought in an annual appropriations bill. Mulvaney said this misunderstanding reflected Schumer's duplicitous negotiating tactics.

"That is not 'all of the money for the wall,' nor was it ever intended to be all the money for the wall," said the budget director, who had a reputation in Congress for being a fiscal conservative. Trump has repeatedly claimed, on the campaign trail and during his presidency, that Mexico would pay for the wall.

Matt House, a spokesman for Schumer, disputed Mulvaney's account.

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Schumer's remarks early Saturday morning followed those of Senate Majority Leader Mitch McConnell, who blamed Democrats for the Senate's failure to approve a bill passed by the Republican-controlled House on Thursday.

Schumer, however, rejected that characterization and pointed out that four Republican senators – Mike Lee of Utah, Jeff Flake of Arizona, Lindsey Graham of South Carolina and Rand Paul of Kentucky – voted in earnest against the continuing resolution. The New York Democrat then turned his ire on the president himself.

"The blame should crash entirely on President Trump's shoulders," Schumer said. "This will be called the Trump shutdown because there is no one who deserves the blame ... more than President Trump."

-CNBC's Christina Wilkie contributed to this report.