WASHINGTON—Senate Democrats are urging Republicans to postpone a final vote on the GOP’s tax overhaul legislation until Democratic Sen.-elect Doug Jones of Alabama takes his newly won seat. But Republican leaders have shown no willingness to do so.

Republican party leaders, in the last stages of finishing the $1.4 trillion tax bill, aim to hold votes next week after a conference committee resolves differences between bills passed by the House and Senate.

“We have a previously announced timeline and if there’s any scheduling changes, I’ll be sure to let you know," Don Stewart, a spokesman for Senate Majority Leader Mitch McConnell (R., Ky.), said in an interview.

Sticking to the timeline they have been using for months would enable President Donald Trump to sign the bill into law before Tuesday night’s election results are officially certified and Mr. Jones is sworn in. Alabama’s chief election official, Secretary of State John Merrill, said on CNN on Wednesday the result would be certified between Dec. 26. and Jan. 3.

“It would be wrong for Senate Republicans to jam through this tax bill without giving the newly elected senator from Alabama the opportunity to cast his vote,” said Senate Minority Leader Charles Schumer (D., N.Y.), a position echoed by other Democratic senators. “Will they have the strength and the political foresight to start over, or at least to modify their bill? Or will they just barrel ahead, pretending not to hear the voices of the middle class crying out against this bill?”

Sen. John Cornyn (R., Texas), the no. 2 Republican in the Senate, said he didn’t expect the Alabama result to have “any real effect” on the tax bill and didn't think Alabama voters were sending a message about it.

“To have a Democrat win in Alabama is pretty extraordinary, and I think it’s all about the flawed candidate,” he said.

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Republicans have a 52-48 edge in the Senate, including temporary Sen. Luther Strange (R., Ala.), whose seat will soon be occupied by Mr. Jones. Mr. Jones will tilt that balance to 51-49. Just one Republican voted against the tax bill earlier this month and Vice President Mike Pence can break ties. That means Mr. Jones wouldn’t, by himself, be able to change the outcome.

But his presence would give new leverage to Republican senators who voted for the tax bill but have expressed concerns about the emerging agreement with the House. That list includes Marco Rubio of Florida, who wants a more generous child tax credit for low-income families, and Susan Collins of Maine, who is concerned about a repeal of the requirement most individuals carry health insurance and discussion of lowering the top individual tax rate, among other issues.

Any three Republican senators could bring the tax bill to a standstill by refusing to vote for it until Mr. Jones is seated. So far, none has said he or she would.

Democrats pointed to their own actions in 2010, after Republican Scott Brown won a special election for a Massachusetts Senate seat. Even though the House and Senate had already each passed their own bills, after the election, then-Sen. Jim Webb (D., Va.) said he wouldn’t vote for health care bills until Mr. Brown was seated.

That statement and comments by Democratic leaders quickly halted any discussion of attempts to reconcile the differences between House and Senate bills and deprived Democrats of the 60-vote majority that let them break Republican filibusters.

Democrats didn’t finish health care legislation between Mr. Brown’s victory and his swearing-in, and eventually used a different procedural path to complete the legislation with a simple majority.

The parallels aren’t perfect. The Massachusetts campaign was much more focused on health care than the Alabama special election was on tax policy.

Write to Richard Rubin at richard.rubin@wsj.com