The political furor over the indictment this week of former President Donald J. Trump over his efforts to overturn the 2020 election is spilling into the escalating congressional spending fight as conservatives, following the former president’s lead, take aim at federal law enforcement agencies, raising yet another obstacle to avoiding a government shutdown.

Representative Majorie Taylor Greene, the far-right Georgia Republican who has become a close ally of Speaker Kevin McCarthy, reacted to the indictment by vowing to try to cut funding for the special counsel Jack Smith while suggesting she would oppose other law enforcement spending measures as well. With Republicans already pressing for deep spending cuts and social policy requirements that have alienated Democrats, they will likely have only four votes in their own party to spare, meaning just a handful of defections could sink the bills.

“This is nothing but a political assassination, and I will not vote to fund a communist regime,” Ms. Greene said in a statement after the latest indictment of Mr. Trump, the front-runner in the Republican presidential primary race. “I will not vote to fund a weaponized government while it politically persecutes not only President Trump but all conservative Americans.”

Her broadside echoed one by Mr. Trump himself, who after pleading not guilty in April to 34 felony charges in Manhattan alleging that he orchestrated a hush-money scheme to pave his path to the presidency and then sought to cover it up, called for cutting funding of the Justice Department on his social media platform.

“REPUBLICANS IN CONGRESS SHOULD DEFUND THE DOJ AND FBI UNTIL THEY COME TO THEIR SENSES,” he wrote then.

Any attempt by the House to do his bidding would be dead on arrival in the Democratic-led Senate and at President Biden’s White House. But the Republican unrest over the indictment appears to have injected a powerful new political incentive into the struggle over spending, increasing Republicans’ appetite for a shutdown fight. That could present a difficult new dynamic for Mr. McCarthy as he seeks to placate the conservative wing of his party while avoiding a lapse in government funding on Oct. 1.

A right-wing advocacy group with significant influence among the most conservative House Republicans has been clamoring for months for deep cuts and “systemic changes” to the F.B.I., an approach that could gain momentum in light of the indictment. Lawmakers who had been hesitant to slash the law enforcement budget may now be emboldened to do so.

“Ending the weaponization of the F.B.I. means defunding the worst areas of corruption & the focus on intelligence that led it away from actual law enforcement,” Russ Vought, a former top Trump administration budget official who now leads the right-wing Center for Renewing America, wrote late last month on X, the social media platform formerly known as Twitter.

Mr. Vought’s group called for more than $2.5 billion in F.B.I. reductions — a nearly 25 percent cut. Those cuts would be far below the reductions House Republicans are already considering in what is already seen as an austere plan.

House Democrats already uniformly oppose the emerging spending bills since they are below the spending levels agreed to by Mr. Biden and Mr. McCarthy in their deal to suspend the federal debt limit and contain numerous conservative policy riders they find objectionable.

Should conservatives prevail in their insistence on even deeper cuts and other restrictions on federal law enforcement, it could drive off more mainstream Republican votes. If conservatives like Ms. Greene do not get what they want and oppose the legislation, Mr. McCarthy would face a painful dilemma: Either allow the spending measures to fail and force a government shutdown for which his party would almost certainly be blamed or cooperate with Democrats to pass the bills and put his leadership position at risk.

The conservative animosity toward the F.B.I. is a stark break with the traditional Republican orthodoxy of strong support for law enforcement. It has little traction in the Senate, where Democrats and Republicans have been working in a bipartisan fashion to advance spending bills for consideration when the Senate returns next month. Most senators of both parties would be opposed to entertaining the sort of spending reductions for law enforcement sought by the House conservatives.

“I do not believe that there will be support in the Senate for defunding the F.B.I. despite its mistakes outlined by the inspector general, nor do I believe that an effort to restrict the Department of Justice would be successful,” said Senator Susan Collins of Maine, the top Republican on the Appropriations Committee, referring a 2019 report on the Justice Department’s investigation into links between Russia and Trump campaign aides in 2016. “Our country is experiencing a crime wave, and we are in the midst of a serious drug epidemic. We need more law enforcement officers, not fewer.”

But members of the Freedom Caucus and other hard-right lawmakers in the House say the Department of Justice’s pursuit of Mr. Trump and those arrested and jailed for the Jan. 6, 2021, assault on the Capitol are examples of how the F.B.I. has lost its way in the aftermath of the Sept. 11, 2001, terror attacks, which resulted in Congress granting them new powers.

“I’d like to take it back to a minimum, sort of pre-9/11 focus on crime and working with state and local jurisdictions to combat crime and not be so much dwelling on domestic terrorism,” said Representative Chip Roy, a Texas Republican and member of the Freedom Caucus.

He and other conservatives are looking to at least block construction of an estimated $4 billion new F.B.I. headquarters in the Washington suburbs, a project being hotly pursued by Democratic lawmakers from Maryland and Virginia.

The Center for Renewing America proposes steps well beyond blocking the headquarters. In a July 25 report, the organization called for significant changes in operational funding for the F.B.I., including eliminating all spending on the intelligence branch, saying the $1.9 billion arm of the F.B.I. had become a domestic intelligence agency turned against “law-abiding Americans and those holding disfavored views.”

House Republicans were already struggling to win approval of their spending measures because of an internal divide over additional spending cuts sought by conservatives and abortion rights restrictions and other policy initiatives opposed by fellow Republicans.

When Congress returns in September, the House and Senate will have just a few weeks to try to pass their spending bills and reconcile their significant differences before the Sept. 30 deadline marking the end of the fiscal year, an outcome that seems highly unlikely.

To avoid a shutdown, Congress would then need to pass a stopgap spending bill, but even that temporary solution is no sure thing, given conservative demands for guarantees of deep spending cuts before moving forward.