WASHINGTON—Not a single member of Congress who represents the territory on the southwest border said they support President Donald Trump’s request for $1.4 billion to begin construction of his promised wall, according to a Wall Street Journal survey, testing the administration’s ability to reach a deal on government funding next week.

Most lawmakers representing the region—both Democrats and Republicans—said they are opposed and many said they have unanswered questions. A few were noncommittal, but not a single member of the House or Senate representing the region expressed support for the funding request. That includes nine members of the House and eight senators across four states: Texas, New Mexico, Arizona and California.

The question is increasingly urgent as Congress and the White House scramble to agree on a spending bill needed to keep the government open. Existing funding for the government expires April 28, and the White House says it wants funding for the border wall as part of the package. Senior congressional Republicans have long indicated that they prefer to leave it out. That is partly because Senate Democrats are opposed, and their votes will be needed because most legislation requires 60 votes to clear the chamber, where Republicans hold 52 seats. Congressional Republicans have said they don’t want to risk the partial government shutdown that such a showdown could trigger.

Still, the White House is holding firm on its position.

“You’re always going to have constituencies within both parties that have local issues—we get that,” Mick Mulvaney, the head of the Office of Management and Budget, said in an interview. He added that the GOP leadership is on board to fight for the funding because “they know it’s a priority for the president.”

A spokesman for House Minority Leader Nancy Pelosi replied by noting that Mr. Trump had promised Mexico would pay for the wall: “The White House’s demands that American taxpayers now foot the bill for a multi-billion dollar boondoggle are intensely opposed by Democrats and many Republicans.”

Enthusiasm for the project has been tepid, at best, from many lawmakers since Mr. Trump submitted his request. In March, he asked Congress for $1.4 billion in spending for the current fiscal year for the project, with an additional $2.6 billion for next year. Many Republicans responded that the Trump approach is overly focused on a physical barrier rather than other approaches to border security, such as technology, that experts say can be more effective and less expensive.

Administration officials said the 2017 money would pay for 48 miles of new border and levee wall systems, and 14 miles of replacement fencing, as well as some technology improvements and road construction.

“As representatives of the communities that make up our southern border, we recognize the need for robust border security and infrastructure to ensure public safety and increase cross border commerce,” Reps. Will Hurd (R., Texas) and Martha McSally (R., Ariz.) wrote in a letter last month to senior administration officials. “However, we also have an obligation to be good stewards of taxpayer dollars and as such have a number of questions.”

Some who back immigration restrictions dismiss these lawmakers’ concerns as parochial and irrelevant to the national imperative to secure the border. But opposition from border-state members is significant in that many know the issue best. Ms. McSally, for example, chairs the Homeland Security border security subcommittee, and in an interview, she said walls do little to stop criminal organizations from getting across the border. “They will go over, through or under physical barriers, sometimes pretty quickly,” she said.

Mr. Hurd, whose district includes 800 miles along the border, describes a wall as “the most expensive and least effective way to secure the border.”

A third House Republican whose district touches the border, Rep. Steve Pearce of New Mexico, also opposes funding if it is focused on a wall, as opposed to a more comprehensive approach. “The solution must be a dynamic, multifaceted one,” he said.

Related Despite enthusiastic backing for President Donald Trump and pleas for a stronger border, Arizona ranchers are conflicted in their support for the promise to build a wall along the border with Mexico. Photo/Video: Jake Nicol/The Wall Street Journal

In the Senate, none of the four Republicans representing border states have expressed support for the project as conceived by Mr. Trump.

Sen. John Cornyn of Texas, the No. 2 Republican in the Senate, has repeatedly voiced concerns about the Trump spending request, including the impact of adding to the debt as well as the wisdom of the project. Mr. Cornyn has long said the Trump wall proposal is too narrowly focused, and he says he is working on legislation that would take a more comprehensive approach to border security.

Sen. Jeff Flake (R., Ariz.) has said he also thinks border security should be focused elsewhere. “I will continue to review options as the current appropriations process moves forward,” he said in a statement Friday.

Sens. Ted Cruz (R., Texas) and John McCain (R., Ariz.) have both voiced skepticism about the Trump plan’s focus on a wall over other border security measures. A Cruz spokesman said the senator backs the overall idea but wouldn’t say whether he backs the supplemental spending request. A McCain spokeswoman declined to comment on the spending request.

Meanwhile, House and Senate Democrats are universally opposed—some forcefully so.

“There is no way in hell I support the request for $1.4 billion in border wall spending. My view on the wall is that we should bulldoze the existing structures,” said Rep. Filemon Vela (D., Texas).

Rep. Raúl Grijalva (D., Ariz.) joined a lawsuit seeking to block the project based on environmental factors. The other House Democrats opposing the project include Reps. Juan Vargas, who represents the entire California section of the border; and Texas Reps. Beto O’Rourke, Henry Cuellar and Vicente Gonzalez.

White House budget director Mick Mulvaney listened as President Donald Trump met with lawmakers at the White House last month. Photo: jonathan ernst/Reuters

“The idea of a wall sounds good as campaign rhetoric, but the campaign is over and we need to offer the American people real solutions, not a false sense of security,” Mr. Gonzalez said.

In the Senate, Democrats Dianne Feinstein and Kamala Harris of California and Tom Udall and Martin Heinrich of New Mexico are all strongly opposed, according to statements and interviews.

One activist who backs tighter immigration laws, Mark Krikorian, argued that border lawmakers’ views fail to take into account the national interest.

“It doesn’t surprise me that congressmen near the border are going to be either opposed or ambivalent but that’s neither here nor there in deciding whether it’s a good idea,” he said.

—Peter Nicholas contributed to this article.

Write to Laura Meckler at laura.meckler@wsj.com and Kristina Peterson at kristina.peterson@wsj.com

Corrections & Amplifications

Sen. Ted Cruz backs the overall idea of a border wall, but his spokesman wouldn’t say whether he supports a supplemental spending request. An earlier version of this article omitted Mr. Cruz’s general support for the concept. (April 23)

The U.S. border wall pictured in this article tops a levee near the Rio Grande at the U.S.-Mexico border near Hidalgo, Texas. An earlier version of the caption accompanying the photo incorrectly said the wall was near La Grulla, Texas. (April 26)