A package of House-passed spending bills stalled in the Senate on Wednesday, as lawmakers appeared to dig in for a battle over President Trump’s U.S.-Mexico border wall ahead of a Sept. 30 funding deadline.

House Democrats introduced separate stopgap legislation that would keep the government running through Nov. 21, with Democratic leaders expressing an interest in kicking the tougher fights over Mr. Trump’s promised border wall until later in the year.

On a mostly party-line vote, the Senate voted 51-44 to move forward on the four-bill package funding programs in defense, labor, health, education, energy, water and foreign operations.

That was nine votes short of the 60 votes needed to overcome a filibuster, though Majority Leader Mitch McConnell voted no for procedural reasons.

The procedural vote had been expected to fail. Had it succeeded, it would have allowed Senate Republicans to substitute their own version of the legislation and push the process closer to final negotiations with their counterparts in the Democrat-led House.

Senate Appropriations Committee Chairman Richard Shelby had said Senate Republicans were planning to substitute their own four bills totaling close to $1 trillion, which would be the lion’s share of the federal government’s discretionary budget for fiscal 2020.

“Our national security is undermined by the failure of Congress to pass the military’s budget on time,” said Mr. Shelby, Alabama Republican. “The fiscal year ends soon, and this is no time for partisan politics.”

But Democrats have warned Mr. Shelby and Mr. McConnell that they will not support spending legislation that diverts funds from other programs to go toward a wall on the southern border.

Democrats are incensed after Mr. Trump told Congress this month that he planned to use his emergency powers to divert about $3.6 billion in Pentagon funds to the wall.

“We’re simply trying to stop Republicans from stealing the money from our military and putting it into the wall, which he said Mexico would pay for,” said Senate Minority Leader Charles E. Schumer.

A dispute over border wall funding helped trigger the 35-day partial government shutdown that started in December and stretched into January.

On the House side of the Capitol, Democrats introduced a separate stopgap funding bill that would keep the government running through Nov. 21, setting the new deadline for right before lawmakers are supposed to break for Thanksgiving.

Congress usually doesn’t pass all of its annual spending bills on time, so lawmakers frequently resort to a stopgap bill, known as a “continuing resolution” or CR in Capitol-speak, to temporarily keep the government running at existing levels and give lawmakers more time to negotiate the bigger bills.

“After we pass this CR and the Senate moves forward with their process, I look forward [to] negotiating responsible bills that uphold our values and give working families a better chance at a better life,” said Appropriations Committee Chairwoman Nita Lowey, New York Democrat.

House Majority Leader Steny H. Hoyer had told reporters earlier in the day that he and House Speaker Nancy Pelosi wanted to keep the bill as “clean” as possible.

Both parties signed off on a two-year deal this summer that will allow Congress to spend about $320 billion more than what’s allowed under current law, but lawmakers still need to write and pass bills that appropriate those funds to avert a potential shutdown.

Mr. Shelby said language in the deal bans restrictions on the president’s ability to transfer funds.

“Our Democratic colleagues may now regret having agreed to these terms, but that does not change the fact that they did,” he said.

Sen. Patrick J. Leahy, the top Democrat on the Senate Appropriations Committee, said he was hopeful that once leaders secured the budget deal they would be on a “bipartisan path” to consider actual spending bills.

“But the Republican leadership started out the process on a partisan note by refusing to rein in the ability of the president to take the defense dollars that we had [supported] to help our troops … to build his wall,” the Vermont Democrat said.

Mr. Leahy did say he would urge the Senate to take up Ms. Lowey’s stopgap bill and that he would try to find a path forward with Mr. Shelby in the broader negotiations.

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