The accusations of Russian interference in the November presidential election have provided a last-minute lifeline for the long-shot effort to head off Donald Trump at the Electoral College.

Ten electors, led by House Minority Leader Nancy Pelosi’s daughter Christine Pelosi, demanded Monday an intelligence briefing on Russian interference by Monday, when the Electoral College makes the vote official.

Flipping the Electoral College seems to be the only hope left for Trump opponents. Recount efforts in Wisconsin and Pennsylvania have backfired.

The Progressive Change Campaign Committee urged Trump opponents to participate in a protest scheduled for all 50 state capitols Monday aimed at persuading electors to cast their ballots for someone other than the Republican real estate mogul.

“The idea that a foreign government has interfered with our elections to undermine their credibility, much less support one of the candidates, indicates a Rubicon has been crossed for our nation, and we need to consider it carefully,” Chris Suprun, a Texas Republican elector, said at a Sunday press conference.

Even before Friday’s news reports, Mr. Suprun had vowed not to cast his ballot for Mr. Trump.

The report cited anonymous CIA sources who said Russian-backed hackers specifically tried to boost the Republican’s candidacy by tapping into Democratic email accounts.

Mr. Trump dismissed the reports as the product of a Democrat-fueled effort to delegitimize his presidency.

“I think it’s ridiculous. I think it’s just another excuse. I don’t believe it,” Mr. Trump said in an interview on “Fox News Sunday.”

With the Electoral College vote just days away, however, organizers of the small yet unprecedented drive to upend the results of the Nov. 8 presidential race seized upon the reports, saying they show foreign influence tainted the election.

The group Hamilton Electors, which is urging both Democratic and Republican electors to unite behind a third candidate, preferably a moderate Republican, said this “is the moment Alexander Hamilton warned us about.”

“We now know that the Russian Government actively worked to elect Donald Trump. The Electoral College is our defense against this attack on our sovereignty,” Hamilton Electors said in an online post. “The Electoral College must be our defense against Russia.”

The bid to upend Mr. Trump’s victory gained momentum the same day that Pennsylvania and Wisconsin certified their results of the Nov. 8 election.

The recount in Wisconsin, initiated and funded by Green Party presidential candidate Jill Stein, showed Mr. Trump picking up 131 votes, which changed the outcome by 0.06 percent. He won the state by about 22,000 votes.

Mr. Trump took to Twitter on Monday evening to repeat his claim that the recount effort was just a fundraising opportunity for Ms. Stein. “The Dems and Green Party can now rest. Scam!” he said.

Meanwhile, a federal judge rejected the Green Party’s request for a recount of paper ballots in Pennsylvania, which Mr. Trump also won. The judge said Ms. Stein’s suspicion of voting machines being susceptible to hacking “borders on the irrational.”

Ms. Stein also filed for a recount in Michigan, which a judge halted last week after three days. Mr. Trump carried the state by fewer than 11,000 votes out of 4.8 million cast.

In their letter to Director of National Intelligence James R. Clapper, the 10 electors said they needed to know whether there are “ongoing investigations into ties between Donald Trump, his campaign or associates, and the Russian government interference in the election.”

“We further require a briefing on all investigative findings, as these matters directly impact the core factors in our deliberations of whether Mr. Trump is fit to serve as President of the United States,” said the letter.

The electors said that will also need Mr. Trump to prove “that he and his staff and advisers did not accept Russian interference, or otherwise collaborate during the campaign, and conclusive disavowal and repudiation of such collaboration and interference going forward.”

None of the electors was likely to vote for Mr. Trump anyway, given that Democratic electors are obligated to vote for Democratic presidential candidate Hillary Clinton. Nine of the 10 signers are Democrats, and Mr. Suprun was the only Republican.

In the letter, the electors justified their demand for information as necessary in order to “fulfill our role,” arguing that the Constitution envisioned the Electoral College not as a rubber stamp but as a “deliberative body that plays a critical role in our system of government — ensuring that the American people elect a president who is constitutionally qualified and fit to serve.”

Twenty-nine states and the District of Columbia require electors to vote for the candidate chosen by the voters or face penalties such as fines.

In Colorado, two electors are challenging a state law that binds them to the popular vote in order to avoid being fined as “faithless electors.” While that lawsuit wouldn’t affect them — Mrs. Clinton won Colorado — it could in theory overturn all state laws binding electors and leave them free to vote for whomever they wish.

However, U.S. District Judge Wiley Daniel denied Monday evening their request to issue an immediate injunction against the law, which would have effectively nullified the law when the Electoral College meets.

Similar lawsuits are pending in California and Washington state.

• S.A. Miller contributed to this report.

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