It might be that the Red Wave failed to materialize but as dawn broke Thursday there was still a possibility that the Republicans could yet end up with control of the Senate as well as the House.

That’s what glimmers in the excruciatingly close count so far in three states — Nevada, Arizona, and Georgia. If the Republicans eke out a win in any two of those, and hold onto their lead in the House, Congress would be theirs.

Though we mightn’t know this until the weekend, when Nevada and Arizona are likely to have finished enough of their vote counts to call their races, Georgia is on its way to a run-off in early December between Senator Warnock, the Democratic incumbent, and Herschel Walker, the Republican.

In Arizona, where some 600,000 ballots remained to be counted, the incumbent Democrat, Senator Kelly, maintained a small but dwindling lead over his Republican challenger, Blake Masters. The race is considered too close to call with about a quarter of the total votes cast still uncounted.

Mr. Kelly’s 2020 special election victory gave Democrats both of Arizona’s Senate seats for the first time in 70 years. It was propelled by the state’s fast-changing demographics and the unpopularity of President Trump there at the time.

This time, the unpopular president, Joe Biden, is from Mr. Kelly’s own party, and the environment looks less favorable for Democrats.

The Arizona race is one of a handful of contests that Republicans targeted in their bid to take control of what is now a 50-50 Senate. 

Mr. Kelly led in the initial results Tuesday, which reflected mail ballots returned ahead of Election Day. Mr. Masters was expected to narrow that lead as ballots cast in person are tabulated. Mail ballots will be counted in the coming days.

In neighboring Nevada, the vote count in the Senate race is delayed by the high proportion of mail-in ballots used in the election. With some 80 percent of the votes tabulated, the Republican former state attorney general, Adam Laxalt, holds a roughly two-point lead over the Democratic incumbent, Senator Catherine Cortez Masto.

The race is considered too early to call, and details about outstanding ballots are unclear. As of Wednesday, when an estimated roughly three-quarters of the votes were counted in races statewide, Republicans were leading their Democratic opponents by single-digit percentage points.

Yet a significant number of mail ballots remain to be counted, and Nevada election officials will count ballots received until Saturday as long as they were postmarked by Election Day. Officials have said there are tens of thousands of ballots that remain to be counted in Las Vegas’ Clark County.

Democrats and Republicans are urging their supporters to be patient while officials continue to count votes. In the 2020 presidential election, Nevada wasn’t called until the Saturday after Election Day.

In the Senate race in Georgia, Messrs. Warnock and Walker will meet in a December 6 runoff after neither reached the general election majority required under state law, ensuring an expensive, bitter fight that could still determine which party controls the Senate going forward.

It will be the second runoff for Mr. Warnock, who first won his seat in a January 5, 2021, special election runoff alongside Democrat Jon Ossoff’s victory in a concurrent Senate runoff. Together, the Georgia seats gave Democrats the narrow majority they are now defending.

Whether Georgia becomes a winner-take-all a second time will depend on the results in Arizona and Nevada, where Republicans are trying to oust Democratic incumbents.

Mr. Walker, 60, sought Wednesday to play up the possibility that Georgia will tilt the chamber one way or the other. “Control of the Senate is likely at stake,” Walker’s campaign wrote in a fundraising push. “We don’t have a moment to waste.”

Mr. Warnock, 53, acknowledged the likelihood of a runoff in the wee hours of Wednesday morning as he urged the supporters who remained at his election night party to gird themselves for more.

“I understand that at this late hour you may be a little tired, but whether it’s later tonight or tomorrow or four weeks from now, we will hear from the people of Georgia,” the senator said.

Mr. Walker tried throughout the general election campaign to frame a referendum on national Democrats, caricaturing Mr. Warnock as a yes-man for Mr. Biden amid sustained inflation. The former college and professional football star campaigns as a cultural and fiscal conservative but has offered a scant policy agenda should he be elected.

Mr. Warnock, in turn, has distanced himself from Biden, styling himself as a pragmatist in a partisan era. Mr. Warnock also framed Mr. Walker, a celebrity athlete making his first bid for public office, as unqualified and unfit for high office.

The four-week sprint also could lure Mr. Trump into the fray. In the final weeks of the campaign, Mr. Trump held rallies in several battleground states, recycling his claims that the 2020 loss was fraudulent and teasing the possibility of a 2024 presidential run.