The House will be joining several other lawsuits already filed. Plaintiffs in another suit challenging Mr. Trump’s border-barrier plans sought on Thursday to expedite obtaining a judicial order to block the government from taking further steps toward building new border barriers — even as administration officials raced to spend what money they had as quickly as they could.

The plaintiffs — the Sierra Club and the Southern Border Communities Coalition, which are represented by the American Civil Liberties Union — filed a motion for a preliminary injunction that would bar the administration from building new barriers near Yuma, Ariz., and El Paso until the litigation can be resolved on the merits. Last month, the Trump administration had identified those sectors as locations for new or replacement barrier projects, while redirecting $1 billion in excess military pay and pension funds for that purpose.

That part of Mr. Trump’s plans relies not on his high-profile invocation of emergency powers, but on another component that has received less attention: using the Pentagon’s separate authority to redirect some of its annual budget in order to beef up its counternarcotics programs, which in turn can be tapped for blocking smuggling along the border. Because that money must be spent before the fiscal year ends on Sept. 30, the administration is trying to move quickly to issue contracts.

But the A.C.L.U. says that its clients are likely to prevail in their argument that it is unlawful for Mr. Trump to use such tactics to spend more on border barriers than Congress appropriated. Its motion maintains that freezing any further steps toward construction before a final ruling — even as the clock continues to tick toward the end of September — would be justified because any such construction would, among other things, “disrupt these ecosystems and cause irreversible damage.”

That case is pending before a Federal District Court judge in Oakland, Calif., Haywood S. Gilliam Jr., who was appointed in 2014 by President Barack Obama. The plaintiffs have asked him to hold a hearing on the request on May 9.