McConnell praised the American intelligence community, saying he has "the highest confidence in the intelligence community, and especially the Central Intelligence Agency" -- which Trump had recently lambasted over its findings.
McConnell's comments were an implicit rebuke of Trump, who has questioned whether Russia actually interfered with the election, including with hacks of Democratic operatives.
A bipartisan group of senators -- John McCain, R-Arizona, Senate Democratic leader Chuck Schumer, Sen. Lindsey Graham, R-South Carolina, and Sen. Jack Reed, D-Rhode Island -- had on Sunday called for an investigation into the intelligence community's finding that Russia attempted to influence the election.
The President-elect on Sunday morning blasted the intelligence community anew, calling its assessment that Russia interfered in the election "ridiculous."
"I think it's just another excuse. I don't believe it," Trump said in an interview on "Fox News Sunday."
McCain, who has called for an investigation into the hacking, told CNN's Jake Tapper Monday on "The Lead" his own 2008 presidential campaign was hacked by the Russians.
"The key to this is to try to find out what the Russians' intentions were. Were they intending to change the outcome of our election? If so, it's more serious," McCain said.
McConnell also expressed skepticism about Russia's role.
"Let me just speak for myself: The Russians are not our friends," McConnell, a Kentucky Republican, told reporters at a Monday morning news conference.
"I think we ought to approach all of these issues on the assumption the Russians do not wish us well," he said.
House Speaker Paul Ryan, R-Wisconsin, took the same position as McConnell.
"Any foreign intervention in our elections is entirely unacceptable," Ryan said in a statement. "And any intervention by Russia is especially problematic because, under President Putin, Russia has been an aggressor that consistently undermines American interests."
Two days earlier Trump sided with Russia over the CIA and attacked the US intelligence assessment of Russia's role.
"These are the same people that said Saddam Hussein had weapons of mass destruction," Trump's transition team said in a terse, unsigned statement targeting the CIA on Friday.
Though he supported calls for an investigation into Russian hacking, McConnell poured cold water on the idea -- which McCain had proposed -- of a select committee to investigate Russian hacking.
Ryan, too, said the House Intelligence Committee, which has investigated cyberhacking, would handle the matter -- rejecting Democrats' calls for a bipartisan joint committee.
Trump, meanwhile, continued dismissing intelligence reports Monday morning, tweeting
, "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?"
Ignoring the conclusions of the intelligence community that Russia sought to meddle in the US election is dangerous, Sen. Angus King said Monday.
The Maine independent, who caucuses with the Democrats in the Senate and sits on the Senate Intelligence Committee, told CNN's Chris Cuomo on "New Day" on Monday that what Director of National Intelligence James Clapper tells you, "you can take it to the bank."
"If you blow off what Jim Clapper's telling you, you do so at your peril," King said.
Clapper is a retired lieutenant general with decades of intelligence experience across administrations of both parties, King pointed out. ODNI pulls together intelligence from all of the US government's intelligence agencies.
Clapper's office, along with the Department of Homeland Security, put out a statement in October declaring that the US government and intelligence had concluded that senior levels of the Russian government had directed the hacking of Democratic political groups in the US and release of information from them with the intent of influencing the US election.
A series of leaks of emails and internal documents from the Democratic National Committee and Hillary Clinton's campaign chairman over the weeks and months leading up to the election jolted the race, vexing Democrats and resulting in the resignation of officials from the DNC including its former chairwoman. Republicans did not suffer the same release.
The Russia involvement has been a point of contention, however. Despite overwhelming consensus between the private sector and intelligence community that Moscow was directing the attacks and involved in the leaks, Trump has continually denied that Russia played a role, saying over the weekend he doesn't believe the conclusion.
The latest dust-up has come over a report
from The Washington Post that the CIA has concluded Russia didn't just seek to sow discord in the election, but actually sought to get Trump elected, following an earlier report
King said while questions remain about the Moscow's intention, there's little doubt about Moscow's involvement.
"I think (Trump) ought to sit down with some of the top people ... and take a deep breath and listen to this," King said. "I think he's too defensive right now, and I can understand that. This is serious business, but you know this is not a partisan issue."
King also supported the efforts of his colleagues McCain, Schumer, Graham and Reed to hold bipartisan hearings on the topic.
McCain and Schumer appeared together on "CBS This Morning" to discuss the need for the hearings. They, too, agreed the evidence is clear Russia engaged in hacking, though the intentions are still under question.
"I can't reach that conclusion yet, which is why we need a bipartisan effort to uncover the whole situation," McCain said. "It's another form of warfare and the entire issue is going to be examined by the Armed Services Committee because it's a threat to our national security."
Schumer said the seriousness of the situation calls for a bipartisan approach that doesn't devolve into politics.
"We need to get to the bottom of this in a fair, nonpartisan, non-finger pointing way," Schumer said. "This is something that should unite Republicans, Democrats, conservatives, liberals, patriots."
The White House strongly suggested Monday that Trump was the beneficiary of Russian meddling in the US election, saying it was the President-elect who praised Vladimir Putin on the campaign trail.
"The emails that had been hacked and leaked by the Russians, these were emails from the DNC and John Podesta, not from the RNC and Stephen Bannon," said White House Press Secretary Josh Earnest.
"It was the President-elect who over the course of the campaign indicated that he thought that President Putin was a strong leader," Earnest added.
He said members of Congress should "spare us the hand-wringing" and move forward with investigating Russian ties to the election.