Fresh signs emerged Sunday that President-elect Donald Trump could embrace the intelligence community’s view that the Russians were behind a computer-hacking operation aimed at influencing the November election.

A senior Trump aide said Mr. Trump could accept Russia’s involvement if there is a unified presentation of evidence from the Federal Bureau of Investigation and other agencies. This followed weeks of skepticism from the president-elect and his supporters that there is sufficient evidence that Russia was responsible for cyberattacks against the Democratic National Committee or leak of stolen emails.

Speaking on Fox News Sunday, Mr. Trump’s incoming chief of staff, Reince Priebus, said the president-elect “would accept the conclusion if these intelligence professionals would get together, put out a report, show the American people that they are actually on the same page.”

Related Video Donald Trump signaled during the campaign that he would take a more cordial stance towards Russia. However, allegations of hacking during the election may hamper negotiations on issues including Ukraine, Syria and arms control. Photo: AP

His statement follows an intensifying bipartisan push on Capitol Hill to launch a separate investigation into the matter. Mr. Trump has called for opening up new lines of cooperation with Russia, and some of his critics in both parties have said his refusal so far to say Russia tried to interfere in the election was a sign that he doesn’t believe that Moscow is a U.S. adversary.

A number of Republicans have advised Mr. Trump to proceed cautiously with Russia, saying the nation’s president, Vladimir Putin, is looking to expand his influence in the Middle East and Eastern Europe and threaten American alliances.

On Sunday, four senators—including Republicans John McCain and Lindsey Graham—sent a letter to Majority Leader Mitch McConnell calling for a temporary select committee “on Cyber” to probe the alleged Russian hacks, among other things. They said that allowing existing congressional committees to look into the hacks is difficult because there are too many committees that have claimed oversight.

“We believe it is justified by the extraordinary scope and scale of the cyber problem,” the letter said, which was also signed by Democratic Sens. Charles Schumer of New York and Jack Reed of Rhode Island. “Democrats and Republicans must work together, and across the jurisdictional lines of the Congress, to address this unique challenge.”

Mr. McConnell last week backed a probe of Russia’s activities by Senate committees but hasn’t endorsed the creation of a new panel.

The lawmakers have said this select committee, which would need authorization from GOP leaders, would be “time-limited” and face certain deadlines. Select committees can have subpoena power, conduct interviews and have access to classified information.

U.S. intelligence officials in October reached an assessment that the Russian government launched a cyberoperation to steal emails and other documents from Democratic Party officials and then selectively leak thousands of records to interfere with the presidential election. The Central Intelligence Agency recently found that Russian officials wanted to help Mr. Trump win, a finding that Mr. Trump and his top aides have angrily dismissed.

A broader group of Republicans have said they aren’t sure that Russia was specifically trying to assist Mr. Trump.

Former Defense Secretary Bob Gates, who served in both the George W. Bush and Obama administrations, said Sunday he still wasn’t convinced that Russians were trying to specifically aid Mr. Trump, but he said their involvement in the operation seems undeniable.

“Maybe part of the problem was that it took the intelligence community a while to assemble really firm evidence of Russian involvement and Russian government involvement that delayed a response,” Mr. Gates said on NBC. “Attribution is a challenge, but it seems pretty clear to me that they have developed really reliable information that the Russian government was involved.”

Mr. Priebus on Sunday suggested the FBI has been slow to accept the CIA’s recent view. But he also suggested that Mr. Trump and top aides were anticipating more information about the alleged Kremlin involvement, following weeks of rejecting that there would ever be a unified consensus.

“We haven’t heard from [FBI Director James] Comey,” Mr. Priebus said Sunday. “These guys should be straight with the American people and come out and say it. I don’t think they’ve been clear about it.”

FBI officials didn’t respond to a request for comment.

The FBI doesn’t make intelligence analysis assessments like the CIA, but the two agencies agree on Russia’s role in the election season hacking of the DNC and Clinton campaign chairman John Podesta, according to people familiar with the matter.

Mr. Priebus said he couldn’t predict how Mr. Trump might respond if the intelligence community and law enforcement presented him with a unified report about the alleged Russian hacking operation. He said, though, that he believed a report on the matter would come eventually.

“I think they will get there, and when they do we can hear from the president-elect and get his opinion,” he said. He said, though, that even if Russia had tried to help get Mr. Trump elected, “there’s no evidence that shows the outcome of the election” changed because of the hack-and-leak campaign.

One week ago, Mr. Trump told Fox News Sunday that intelligence officials didn’t know who carried out the cyberoperation, contradicting public statements made by the Director of National Intelligence and the Department of Homeland Security on the matter.

“They have no idea if it’s Russia or China or somebody. It could be somebody sitting in a bed some place,” Mr. Trump said on Fox News Sunday on Dec. 11. “I mean, they have no idea.”

Since then, Mr. Trump and his aides have repeatedly tried to debunk any alleged connection between the hacking operation and his victory. His closest aides have said the current debate about the hack is an effort to “delegitimize” his victory.

When reports first surfaced more than a week ago that CIA officials believed the Russians were trying to help Mr. Trump, his transition team published a short statement that took aim that the intelligence agency’s credibility.

“These are the same people that said Saddam Hussein had weapons of mass destruction,” the transition team said at the time, referring to a conclusion that turned out to be wrong.

President Barack Obama said at a new conference Friday the U.S. government would release some information about the matter but that there would be a limit to what is disclosed.

“We will provide evidence that we can safely provide, that does not compromise sources and methods,” Mr. Obama said. “But I’ll be honest with you, when you are talking about cybersecurity, a lot of it is classified and we’re not going to provide it, because the way we catch folks is by knowing certain things about them that they may not want us to know.”

Russian officials have denied being behind the attacks, though Mr. Putin has said whoever carried out the campaign conducted a public service by exposing internal Democratic Party emails.

The Office of the Director of National Intelligence is preparing a report on the campaign and is expected to make some of its findings public before Mr. Obama’s term ends on Jan. 20.

—Devlin Barrett contributed to this article.

Write to Damian Paletta at damian.paletta@wsj.com and Kate O’Keeffe at kathryn.okeeffe@wsj.com