Donald Trump appeared poised to sew up his victory Monday despite opposition groups’ efforts to block his path by persuading members of the Electoral College to snub him.

The presidency is awarded to the candidate who wins at least 270 votes in the Electoral College, with each state assigned a certain number of votes based on population. Mr. Trump won a majority of the popular vote in enough states to accumulate 306 electoral votes to Hillary Clinton’s 232.

Liberal groups have urged Trump electors to vote for Mrs. Clinton, another candidate or not at all. But Republicans have said they may lose just one pledged elector, giving Mr. Trump more than enough votes to secure the presidency.

The Associated Press reported Thursday that, based on interviews with more than 330 electors of both parties, there was little evidence of a revolt.

“We expect everything to fall in line,” Reince Priebus, Mr. Trump’s incoming chief of staff, said Sunday on Fox News. He added: “Everything is going to be very smooth” on Monday.

Mr. Trump heads toward his Jan. 20 inauguration with nearly seven in 10 American adults—68%—believing he will change the way business is done in Washington, according to a new Wall Street Journal/NBC News poll. But one-third of that group, or about 20% of all adults, believe it will be the wrong kind of change.

Half of Americans said in the poll they approve of how Mr. Trump is handling his transition, while 41% disapprove. That isn’t nearly the honeymoon President Barack Obama enjoyed. At this point in 2008, 73% said they approved of how he was preparing to become president while 13% disapproved.

The survey also found a majority of Americans are significantly bothered by reports of Russian hacking aimed at influencing the election.

In the run-up to Monday’s Electoral College vote some GOP electors have been inundated with calls, emails and letters from people urging them not to back Mr. Trump.

Brian Westrate, a Wisconsin Republican elector who plans to vote for Mr. Trump, tweeted a photo of more than 600 letters he said he received on Friday alone. Still, he said he respected the efforts. “I fully embrace all of these peoples’ desires to be heard,” he said Sunday, adding the correspondence had been overwhelmingly “respectful and heartfelt.”

Randy Evans, a Republican elector from Georgia, called the efforts “a well-funded, well-organized operation” to deny Mr. Trump the presidency. Mr. Evans, who said he plans to vote for Mr. Trump, “in accordance with the vote of Georgia,” said he has received about 12,000 emails and “I don’t know how many inches thick of letters just yesterday.”

Alex Triantafilou, who heads the Republican Party in Hamilton County, Ohio, said he, too, plans to back Mr. Trump despite being swamped with emails and letters from people trying to sway his vote. He texted pictures showing a box of letters left on his front stoop.

“We have received literally thousands of emails and, gosh, the letters are preposterous,” he said in an interview Sunday.

A number of Democrats pushed for a delay in the Electoral College vote, saying more information is needed about alleged Russian interference in the election, but Republicans such as Pennsylvania elector Bob Asher disagree.

“I don’t think people voted one way or another because of Russia,” said Mr. Asher, who will vote for Mr. Trump despite backing Florida Sen. Marco Rubio in the GOP primaries. “A lot of the public was fed up with the way our nation was going and decided they wanted a change.”

Bret Chiafalo, a Democratic elector from Washington state who co-founded the anti-Trump group Hamilton Electors, said Sunday he remained hopeful about Monday’s vote. “I think enough Republican electors are going to put country before party,” he said.

While the group concluded early on it would be too difficult to get GOP electors to support Mrs. Clinton, they thought there was a chance to persuade enough of them to choose another Republican, such as Ohio Gov. John Kasich, to make it impossible for any candidate to get the required 270 votes.

On Sunday, the group issued a series of tweets invoking previous Republican presidents’ quotes about responsibility with captions such as “GOP Electors: President Abraham Lincoln is speaking DIRECTLY to you.” They are also holding a series of “Unity Vigils” across the country, including one at the Lincoln statue in Spokane, Wash., called “Only the Electoral College can save us from “The Donald.”’

For many of Mrs. Clinton’s supporters, the Electoral College was seen as their last chance to prevent Mr. Trump from becoming president.

Since the Electoral College’s creation in 1787 there have been 157 “faithless electors”—those who didn’t vote for the candidate chosen by their state. Almost half of those votes stemmed from the candidate having died before the Electoral College held its vote, and the rebels have never changed the outcome of an election.

Bret Chiafalo, a Democratic elector from Washington state who co-founded the anti-Trump group Hamilton Electors, said Sunday he remained hopeful about Monday’s Electoral College vote. “I think enough Republican electors are going to put country before party,” he said. Photo: Elaine Thompson/Associated Press

Write to Kate O’Keeffe at kathryn.o’keeffe@wsj.com, Peter Nicholas at peter.nicholas@wsj.com and Damian Paletta at damian.paletta@wsj.com