The White House raised transition tensions with President-elect Donald Trump on Monday, laying out “objective facts” of the mogul’s ties to Russia and lashing out at congressional Republicans who now want to investigate whether Moscow’s cyberattacks were aimed at helping Mr. Trump win the election.

During the campaign, Hillary Clinton’s team charged that Russia was trying to boost Mr. Trump by hacking email accounts and turning the messages over to WikiLeaks, including some that showed the Democratic National Committee plotted to undermine primary rival Bernard Sanders.

“The president-elect didn’t call it into question,” White House press secretary Josh Earnest said of pre-election reports of foreign hacking. Instead, Mr. Trump “called on Russia to hack into his opponent. He certainly had a pretty good sense of whose side this activity was coming down on.”

Mr. Earnest did not outline what the Obama administration did to protect the candidates and the election process.

Senate Minority Leader Harry Reid, Nevada Democrat, piled on by saying he accepted the election outcome but that Russia was actively trying to swing the contest in Mr. Trump’s favor. He also said FBI Director James B. Comey should have sounded the alarm.

“My opinion is yes,” the Kremlin was interfering, Mr. Reid told CNN. “And we got no [information] from the FBI. They ignored it.”

Senate Majority Leader Mitch McConnell, Kentucky Republican, kicked off the day by calling for a bipartisan investigation into foreign interference in the U.S. election. “The Russians are not our friends,” he said.

Mr. McConnell said Sen. Richard Burr, North Carolina Republican and chairman of the Senate Select Committee on Intelligence, will lead the investigation with input from incoming Minority Leader Charles E. Schumer, New York Democrat. Senate Armed Services Committee Chairman John McCain, Arizona Republican, will examine cyberattacks as they relate to warfare.

These endeavors build on President Obama’s call, before he leaves office Jan. 20, for an intelligence report on suspected Russian meddling.

“Obviously, any foreign breach of our cybersecurity measures is disturbing, and I strongly condemn any such efforts,” Mr. McConnell said.

No hard evidence has emerged that Russian intelligence corrupted U.S. election results, despite media reports based on a classified CIA assessment outlining the agency’s belief that Moscow deliberately tried to help Mr. Trump win the presidency. Congressional Democrats quickly seized on the claims.

The CIA on Monday declined to comment on the details of the assessment’s findings.

Mr. Trump lashed out at Democrats highlighting the bombshell report that The Washington Post published Friday. The report, based on anonymous sources within the CIA, said agents concluded that Russian President Vladimir Putin orchestrated cyberattacks to help Mr. Trump’s campaign.

“Can you imagine if the election results were the opposite and WE tried to play the Russia/CIA card. It would be called conspiracy theory!” Mr. Trump said Monday on Twitter.

He later added, “Unless you catch ‘hackers’ in the act, it is very hard to determine who was doing the hacking. Why wasn’t this brought up before election?”

Mr. McConnell said he had “the highest confidence in the intelligence community,” particularly the CIA.

The agency likely went though “a great deal of agonizing” when compiling its assessment over fear that its apolitical reputation would be called into question, said Aki Peritz, a former CIA counterterrorism analyst.

“It is the third rail for the agency,” Mr. Peritz said regarding the political nature of the CIA’s review of Russia’s role. Intelligence agents, officers and officials are “very, very uncomfortable [dealing with] specifically political issues.”

That said, the agency’s findings — if confirmed — are in line with Moscow’s behavior elsewhere in the world.

Russian hackers were able to manipulate vote counts during a 2014 election in Ukraine after they gained access to the country’s electronic balloting system.

Open-source reports also say Russian intelligence officials gained access to reams of U.S. voter databases in Arizona and elsewhere in the U.S. by hacking into state political party and government networks, said Mr. Peritz, though he reiterated that no evidence shows that information was used to help Mr. Trump.

Congressional Republicans and Mr. Trump’s backers have pointed out the agency’s hypocrisy in alleging Russian manipulation in the U.S. election, particularly with the CIA’s history of “covert political action” to prevent communist leaders from taking power during the height of the Cold War.

The agency gained infamy in the late 1970s when covert political action to disrupt national elections in Chile in 1964 and in 1970 were brought to light as part of the congressionally mandated Church and Pike committees created to oversee CIA operations.

“One of the first things [the CIA] did was interfere with the [national] election in 1948 in Italy,” Mr. Peritz said. But U.S. intelligence officials would never “be so cavalier as to break into one side’s [networks] and blast the information all over the internet.”

Democratic presidential candidate Hillary Clinton’s campaign regularly accused Russian-led hacks of breaking into the Democratic National Committee’s networks.

Mr. Earnest said the hacked and leaked information consisted of emails from the DNC and Clinton campaign chairman John Podesta, and not “from the [Republican National Committee] and Steve Bannon.” Mr. Bannon is a top adviser to Mr. Trump.

He also cited the business ties of former Trump campaign chairman Paul Manafort to Russia, Mr. Trump’s praise of Mr. Putin as a strong leader who could help the U.S. in the war against Islamist terrorists and Mr. Trump’s refusal to disclose any business ties with Russia.

“This is all material that was known by Republican politicians in the Congress that endorsed the president-elect,” Mr. Earnest said. “How they reconcile their political strategy and their patriotism is something they’re going to have to explain.”

Mr. McConnell called to investigate foreign meddling through the committee process, or “regular order.” He refused calls from Senate Democrats and House Minority Leader Nancy Pelosi, California Democrat, for an independent, nonpartisan commission to look into suspected Russian interference.

The Republican leader was careful not to criticize the president-elect and said people should wait and see whom Mr. Trump nominates as secretary of state. Reports suggest it will be Exxon Mobil CEO Rex W. Tillerson, who has close ties to Mr. Putin.

“I’m going to save us a lot of time by saying I just addressed how I feel about the Russians,” he told reporters. “And I hope that those who are going to be in positions of responsibility in the new administration share my view.”

House Speaker Paul D. Ryan, Wisconsin Republican, said foreign intervention into U.S. elections was “entirely unacceptable,” though he urged Americans not to use the hacking reports to cast doubt on the “clear and decisive outcome.”

“Any intervention by Russia is especially problematic because, under President Putin, Russia has been an aggressor that consistently undermines American interests,” said Mr. Ryan. “At the same time, exploiting the work of our intelligence community for partisan purposes does a grave disservice to those professionals and potentially jeopardizes our national security.”

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