Lawmakers will return to Washington next week with the main goal of passing a spending bill before an Oct. 1 deadline to keep the government open, but the House’s hardline conservatives want concessions that could dash hopes for a drama-free, election-year spending deal.

Despite repeated promises from House Republican leaders, they again failed to pass all 12 annual spending bills and are homing in on the customary funding patch to get past the Nov. 5 elections.

The House Freedom Caucus wants Speaker Mike Johnson to attach legislation to the short-term spending package that would require voters to provide proof of citizenship.

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Most of the roughly 40 members of the Freedom Caucus said they would vote against a stopgap spending bill if it does not include the citizenship test in the Safeguard American Voter Eligibility Act, known as the SAVE Act, which the House passed this summer with the aid of a handful of House Democrats.

Most Democrats and the White House opposed it.

If Mr. Johnson, Louisiana Republican, relents to the Freedom Caucus demands, the funding patch is not guaranteed to pass in the House, and it is sure to fail in the Democrat-run Senate. That scenario could make the possibility of a politically costly government shutdown a reality.

House Judiciary Chair Jim Jordan, the Ohio Republican who in 2015 was the first chairman of the Freedom Caucus, said attaching a politically divisive measure to a funding patch just over 60 days before Election Day was the right thing to do.

“I don’t want to shut down. Nobody wants to shut down,” he told The Washington Times.

There has never been a shutdown during a presidential election, and Mr. Johnson is unlikely to make history with such a gamble. And most lawmakers are not hungry for a protracted spending fight in September, a crucial time when they could be on the campaign trail.

Lawmakers are set to leave Washington for another long break on Sept. 27, meaning Mr. Johnson may move quickly on a spending patch, but nothing has been set in stone.

Matt Glassman, a senior fellow at the Government Affairs Institute at Georgetown University, said House speakers before Mr. Johnson have had to deal with similar dynamics to keep the government open during election years.

“People like [former House speakers Nancy] Pelosi and [John] Boehner and Johnson, know that the people they got to protect are not in the Freedom Caucus, those are not the people who are about to lose elections,” Mr. Glassman said.

Mr. Johnson recently said he was looking at various options for a stopgap bill and that attaching the SAVE Act was “a big part of this conversation.”

A clear-cut plan has not been presented to lawmakers, and the speaker has not publicly committed to attaching the legislation to a stopgap bill, known on Capitol Hill as a continuing resolution, or CR.

Part of the negotiations include stretching a stopgap bill into next year, which conservatives support, in the hope that former President Donald Trump wins the White House.

While the SAVE Act is popular among Republicans, most don’t want to rock the boat ahead of the elections when control of the House, Senate and White House are at stake.

“I think that any type of grievance within the House of Representatives over a CR or messaging bill, or any type of legislation is not going to help President Trump,” said Rep. Max Miller, Ohio Republican. “And right now, we need to be focused on keeping the government open, which means passing a CR and making sure President Trump wins in November.”