House Homeland Security Committee Chairman Bennie Thompson, D-Miss., broke with some of his fellow congressional Democrats on Sunday by acknowledging in an interview that he "would not rule out a wall in certain instances," although he cautioned that the White House needs a better "plan" than simply using a wall as a "talking point."

However, House Speaker Nancy Pelosi, who preemptively rejected Trump's proposed border wall compromise on Saturday as a "nonstarter," has pledged not to fund the president's border wall under any circumstances -- and, Thompson suggested in an interview with ABC News' "This Week," Trump is stuck with her.

"Mr. President, Democrats will work with you," Thompson said. "But you can't pick what Democrats you work with. We have picked our leaders, and you have to work with our leaders."

Thompson continued, "The notion that we can’t have barriers is just something that’s not true."

Thompson maintained that Democratic leaders remain open to negotiation, one day after President Trump unveiled a plan to offer several immigration-related concessions to Democrats in order to obtain $5.7 billion in funding for a wall on the southern border and end the ongoing partial federal government shutdown, now in its 30th day.

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"I'm saying that we will sit with the chief, the president, whomever, and we will work through it," Thompson told anchor Martha Raddatz. Vice President Pence, speaking to "Fox News Sunday," said that the White House "of course remained open to negotiations."

WATCH: PENCE SLAMS BUZZFEED, TALKS WH PLAN FOR RESOLVING SHUTDOWN IN FOX NEWS INTERVIEW

Pelosi said earlier this month that "a wall, in my view, is an immorality" and "the least effective way to protect the border and the most costly."

That view has the support of the party's progressive wing. In an interview last week, for example, Democratic Texas Rep. Veronica Escobar told CNN that "we know walls don't work, that they don't stop drugs, that they don't stop migration."

Escobar also suggested she was open to destroying existing walls, remarking that they are "really ugly" and "monuments to division."

However, House Majority Leader Steny Hoyer, D-Md., appeared to side with Thompson's position Wednesday on Fox News' "Special Report with Bret Baier," arguing that border walls "obviously" work in some areas and rejecting suggestions that barriers should be removed where they already exist.

The No. 2 House Democrat additionally asserted that the question of whether to fund President Trump's proposed border wall -- a dispute at the center of the unprecedented partial federal government shutdown -- is "not an issue of morality."

But Hoyer, pressed by Baier, distanced himself from those comments.

"Obviously, they work some places," Hoyer responded, when asked if he favored removing border walls. "But the president wanted to first build a wall apparently 1,954 miles of -- and he changed that very substantially. "

TUCKER CARLSON: DEMS BELIEVE WALLS ARE IMMORAL -- AND THAT GOD AGREES WITH THEM

He added: "A wall is -- that protects people is not immoral. I think the issue is whether it works. ... And the debate ought to be not on morality or racism, I will -- I will say that we’re not pleased with some rhetoric that has come about dealing with those -- coming across the border, and we think some of the rhetoric was in fact racist. We think some of that rhetoric was to inflame and was not based upon facts."

Similarly, Thompson on Sunday questioned whether Trump's funding would go toward an effective wall.

FILE - In this Saturday, Jan. 6, 2018, file photo, Texas Democratic Congressman Beto O'Rourke, right, passes a "No Border Wall" sign during a visit to the National Butterfly Center in Mission, Texas, a possible location for a border wall. President Donald Trump is not giving up on his demands for $5.7 billion to build a wall along the U.S.-Mexico border, saying a physical barrier is central to any strategy for addressing the security and humanitarian crisis at the southern border. Democrats argue that funding the construction of a steel barrier along roughly 234 miles will not solve the problems. (AP Photo/Eric Gay, File)

"The president initially started with is not where he is now, and so we don’t know where he will be tomorrow. ... Clearly, Democrats are for border security," he said. "But we are not for this constantly moving the ball just for a talking point."

Democratic leaders have previously supported building border walls. Senate Minority Leader Chuck Schumer, D-N.Y., and other Democrats, including then-Sens. Barack Obama and Hillary Clinton, supported the Secure Fence Act of 2006, which authorized the construction of some 700 miles of fencing at the border. As of 2015, virtually all of that fencing had been completed, according to government figures.

Border Patrol officials have also said that walls are effective. ("We certainly do need a wall. Talk to any border agent and they will tell you that," Border Patrol Chief Carla Provost said on Fox News' "Your World" last month.)

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"The notion that we can’t have barriers is just something that’s not true."
— House Homeland Security Committee chairman Bennie Thompson, D-Miss.

Public ressure to resolve the shutdown one way or the other has mounted in recent weeks, as increasing numbers of airport security officials call out sick and worries grow as to how tax returns and food stamps will be processed.

Trump last week signed a bill guaranteeing back pay for federal workers who have been furloughed or forced to work without pay during the shutdown, which started Dec. 22. The workers are set to miss their next full paycheck on Friday.

Meanwhile, Fox News has learned that Senate Republicans scrambled on Saturday evening to put together legislation tracking President Trump's compromise proposal to end the ongoing partial federal government shutdown, and the text of a bill should be ready on Monday.

Senate Majority Leader Mitch McConnell, R-Ky., indicated to Republican senators on Saturday that he would try to move to start debate on the bill Tuesday.

Tacked onto the legislation will be supplemental money for disaster relief and extensions on the Violence Against Women Act, Fox News has also learned.

In a televised White House address on Saturday afternoon, Trump offered Democrats a three-year extension of protections for 700,000 Deferred Action for Childhood Arrivals (DACA) recipients, in exchange for the $5.7 billion he has been seeking for a barrier along the nation's southern border with Mexico.

The offered deal would also extend protections for 300,000 recipients of the Temporary Protected Status (TPS) program -- which protects immigrants from designated countries with conditions that prevent nationals from returning safely.

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Procedural hurdles would likely prevent the Senate from taking immediate action on legislation that implements Trump's proposal, however. The motion to proceed to start debate on the bill would require 60 yeas. If Democrats don’t cooperate, McConnell will need to file a cloture motion on Tuesday to cut off debate on the motion to proceed, which would not ripen for a vote until Thursday. Republicans hold a 53-47 majority in the Senate.

In his interview with "Fox News Sunday," Pence, asked whether Republicans have the 60 votes to break a potential filibuster, responded: "As the president often says, 'We'll see."

Fox News' Bret Baier, Brooke Singman, Chad Pergram, and Samuel Chamberlain contributed to this report.