Congress reconvenes on Monday after a lengthy summer break for what is shaping up as a pitched three-week battle over spending with significant consequences for both the November election and the next occupant of the White House.

House Republicans have named a politically charged price for agreeing to continue funding the government — legislation that would require proof of citizenship to register to vote. It is their latest salvo in a monthslong effort to push the unsubstantiated idea that a swarm of undocumented immigrants is poised to vote illegally and swing the election to Democrats.

But with a Sept. 30 shutdown deadline looming and a presidential contest on the line, lawmakers also are haggling over the calendar itself. Namely, how long to extend funding and whether the current Congress and President Biden or those elected in November should be responsible for setting spending levels in 2025 and beyond.

Speaker Mike Johnson is preparing to bring to the House floor as early as Wednesday a six-month extension of federal spending through March 28, along with the bill imposing new rules on voter registration. Mr. Johnson, who said in a statement that it was the role of Congress to “ensure that only American citizens can decide American elections,” hopes to draw Senate Democrats into a fight over the demand for new voter registration requirements, despite scant evidence that noncitizens vote.

Senate Democrats are not likely to swallow that provision, which they call unnecessary since federal law already bars noncitizens from voting and driven by xenophobia. In addition, Senate Democrats and some Republicans on both sides of the Capitol would prefer a shorter time span for the interim spending bill, extending funding until the end of December as has been done in similar situations in the past. That would allow Congress to wrap up this year’s appropriations fight by the end of the year and let the next president start with a clean slate.

“I do not support having the C.R. go beyond December,” said Senator Susan Collins of Maine, the top Republican on the Senate Appropriations Committee, using shorthand for the interim spending bill known officially as a continuing resolution. “We are going to have a new administration regardless, and they should be able to concentrate on the new budget year, not have to deal with issues involving the fiscal year that starts Oct. 1.”