Despite no evidence of fraud and a widespread belief that the entire effort will change nothing, Wisconsin election officials this week will begin a statewide recount of the presidential vote, adding yet another chapter to the fight between Donald Trump and Hillary Clinton.

Wisconsin officials will hold a news conference Monday to discuss details of the recount, spearheaded by Green Party presidential nominee Jill Stein.

Ms. Stein has raised about $7 million for the process and also is pressing for recounts in Pennsylvania and Michigan. Top officials with the Clinton campaign have joined the effort.

Mr. Trump won Wisconsin by 27,257 votes and became the first Republican to capture the state in more than 20 years. He garnered 1,404,000 votes to 1,381,823 for the former first lady, according to the Wisconsin Election Commission.

The Trump campaign has dismissed the recount as little more than a publicity stunt. Trump supporters mocked the Clinton campaign Sunday for taking part.

“This is ridiculous. This is a fundraising, notoriety-driven fraud,” Reince Priebus, former head of the Republican National Committee and incoming chief of staff for Mr. Trump, said on “Fox News Sunday.” “I think the American people know this is a waste of everyone’s time and money. … We will win again for the second time.”

Mr. Trump himself called the entire recount bid a “scam by the Green Party” to raise money and to boost Ms. Stein’s political profile.

The president-elect took to Twitter later Sunday to allege, without specific evidence, that there was also “serious voter fraud in Virginia, New Hampshire and California” and accused the media of bias and cover-up on the matter. Earlier Sunday, he said “millions” of illegal votes were cast for Mrs. Clinton.

Despite those objections, Wisconsin officials say the recount will begin by week’s end and that Ms. Stein and her Green Party will pick up the tab. The recount is scheduled to be completed no later than Dec. 13, officials said.

“We have assembled an internal team to direct the recount, we have been in close consultation with our county clerk partners, and have arranged for legal representation by the Wisconsin Department of Justice,” Wisconsin Elections Commission Administrator Michael Haas said in a statement over the weekend. “We plan to hold a teleconference meeting for county clerks next week and anticipate the recount will begin late in the week after the Stein campaign has paid the recount fee, which we are still calculating.”

Wisconsin has found no evidence of voting machine tampering or other foul play, state officials told The Washington Times.

“No evidence of hacking,” commission spokesman Reid Magney said in an email.

Clinton campaign officials also acknowledge that, while they are agreeing to take part in the process, they, too, have no proof of hacking, vote tampering or other interference in the Wisconsin process.

“Because we had not uncovered any actionable evidence of hacking or outside attempts to alter the voting technology, we had not planned to exercise this option ourselves, but now that a recount has been initiated in Wisconsin, we intend to participate in order to ensure the process proceeds in a manner that is fair to all sides,” Clinton campaign attorney Marc Elias wrote in an op-ed for Medium over the weekend.

Absent any evidence that the Wisconsin vote was flawed, some leading voices on the left are downplaying the effort and saying it’s unlikely to change the results of the Nov. 8 presidential election, in which Mr. Trump defeated Mrs. Clinton handily in the Electoral College tally.

Speaking on CNN’s “State of the Union” program on Sunday, Sen. Bernard Sanders — Mrs. Clinton’s opponent in the Democratic presidential primary — said the recount is simply not a big deal.

“It’s taking place. The Green Party has the legal right to do it. … No one expects there to be profound change, but there’s nothing wrong with going through the process,” said Mr. Sanders, dismissing any notion that the result in Wisconsin could change.

“This is exactly the issue — nobody cares,” he said. “I don’t think Hillary Clinton, who got 2 million more votes than Mr. Trump in the popular election, thinks it’s going to transform the election. But do people have the legal right to do it? Yeah, we do.”

The Clinton campaign says it is also open to recounts in Michigan and Pennsylvania, where Mr. Trump also eked out narrow wins. The campaign suggested that it will not initiate recounts in those states but will participate if Ms. Stein or another figure takes the lead.

Mr. Trump’s team also pointed out the irony of Mrs. Clinton’s repeated bashing of her opponent on the campaign trail for refusing to say whether he would accept the election results and now contesting the outcome herself.

“She conceded to him on election night,” Trump campaign manager Kellyanne Conway said Sunday on CNN’s “State of the Union” program. “The idea we’re going to drag this out … is pretty incredible.”

⦁ Rowan Scarborough contributed to this report.

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