(CNN) Top Republicans and Democrats are quietly maneuvering to force a vote on the Senate floor establishing a new committee to investigate the Russian cyberattacks on the US political system, a move that could put GOP leaders who oppose the idea in an awkward spot just as Donald Trump comes into power.

But in a sign of the hurdles ahead for the proponents of the new panel, two key swing votes -- Republican Sens. Jeff Flake of Arizona and Susan Collins of Maine -- are pushing back on the bipartisan efforts to create the committee.

"The Senate Intelligence Committee (along with the Intelligence Committee in the House) is best positioned to continue our investigations into the increasingly brazen and aggressive cyberwarfare emanating from Russia and China," Collins told CNN in an email Monday.

"We don't need to set up a special committee to do what we (can do) through regular order," Senate Majority Leader Mitch McConnell said Monday night on KET in Kentucky.

The procedural fight is significant because the creation of a select committee could raise the public visibility over the Russian involvement in the 2016 elections -- an issue that Trump has repeatedly downplayed.

The effort could play out on the Senate floor in the opening days of the new Congress. Under one proposal being actively discussed, senior senators may try to insert language to create the panel into the organizing resolution for the new Senate, several sources said. If not, the proponents of the committee may insist on a vote in the full Senate on a stand-alone proposal, something that would require 60 votes to overcome any filibuster attempt.

Proponents of the measure are confident that support will grow for the panel as the public learns more about Russia's apparent involvement to meddle in the elections, something many view as a direct attack on the United States. A number of Republicans seem open to the idea of creating a panel, sources told CNN, even as McConnell and House Speaker Paul Ryan, R-Wisconsin, are resistant to the idea.

Advocates for the Select Committee on Cyber say it's needed because the complex issues related to cyberattacks cross the jurisdictions of at least five Senate committees, arguing the best way to understand the problem and find solutions is to bring top senators from each of those committees onto one panel with a clear, if temporary, mandate.

"Cyber is the rare kind of all-encompassing challenge for which the Congress's jurisdictional boundaries are an impediment to sufficient oversight and legislative action," said a letter Sunday to McConnell from Senate Armed Services Committee Chairman John McCain, incoming Democratic Leader Chuck Schumer of New York, Sen. Lindsey Graham, R-South Carolina, and Sen. Jack Reed of Rhode Island, the top Democrat on Armed Services.

The proposal could be tricky for McConnell and Ryan, who both are sharp critics of Russia but now will have to work closely with a new president who has praised Russian President Vladimir Putin and dismissed allegations of Russian hacking. The two leaders say they want existing oversight committees -- spearheaded by the intelligence panels in each chamber -- to examine the issue and possibly propose reforms through legislation or by other means.

Flake, a member of the Senate Foreign Relations Committee, seems to be breaking with McCain, a fellow Arizona Republican.

"Sen. Flake believes that the Senate's standing committees are capable of handling the investigation," said Flake spokesman Jason Samuels.

Collins, the moderate Maine Republican, added: "If we were to create a cyber select committee, it would be months before the committee could even begin its investigation," pointing to the need to hire staff who need to undergo extensive background checks to get security clearances.

"The Senate Intelligence Committee, however, is already staffed with experts and is continuing its investigation," she said.

By keeping the investigations with those top secret committees, the party leaders will have greater control over the directions of the probes and the public release of findings. And Democrats are worried that the secretive Senate Intelligence Committee, led by Republican Sen. Richard Burr of North Carolina, will ultimately bury the findings in a classified setting.

A select committee, empowered with the authority to issue subpoenas, would likely bring significantly more public attention and focus to the Russian hacking issue, which many Democrats blame as a key reason their candidate Hillary Clinton lost the White House to Trump. Each move by the committee -- news conferences, interviews, hearings, findings and recommendations -- would be closely watched and could provide Democratic members a highly visible platform to claim the Russians helped elect Trump.

Select committees have a long history in the Senate, where dozens have been formed over the years -- from Iran-Contra and Watergate in the last century to an investigation into Ku Klux Klan activities in the century before that.

Asked last week if he would back a special committee, McConnell said instead: "We're going to follow a regular order, it's an important subject, and we intend to do it on a bipartisan basis."

While McConnell is rejecting calls to create the committee, the two GOP senators -- McCain and Graham -- are looking to drum up enough support within the Senate Republican Conference to overrule the powerful leader in the opening days of the new Congress. Behind the scenes, Graham and McCain plan to begin courting their colleagues in the coming days to back the idea of a committee, sources said.

And some seem open to the idea. Sen. Tim Scott, R-South Carolina, believes "there should be a full committee dedicated to investigating cyber threats and crimes," according to a spokeswoman for the senator. And Sen. James Lankford of Oklahoma is non-committal, an aide said.

Sen. Cory Gardner, R-Colo., plans to renew his months-long push to create a stand-alone committee to investigate cyberattacks by introducing legislation on the issue next month, his spokesman said.

Several other Republicans who have been tough on Russia didn't respond to inquiries Monday if they would back such a panel -- a sign that opposing the creation of the committee could be a politically difficult vote.

To enact a panel, the Senate would have to pass a resolution that could be blocked by a filibuster in the chamber. That means if the 48 Democrats stay united, they would need at least 12 Republicans to back the proposal.

McConnell would need to be persuaded to allow the vote from coming forward, but Schumer, McCain, Graham and Reed would have leverage to force a vote by blocking other measures on the floor until their demands are met.

The first vehicle next year will be the organizing resolution that the Senate must adopt to establish all of its new committees for the new Congress. In that resolution, provisions could be included to establish the new select committee investigating Russian hacking.

If such provisions are not included, Schumer could presumably seek to block the resolution, potentially throwing a wrench into the opening of the new Senate, including confirmation proceedings for Trump Cabinet nominees.

Sources said tactics for forcing McConnell to act have not been fully discussed yet. And the proponents said they hope a bipartisan deal could be reached to avert a partisan standoff on the floor.

But in the letter to McConnell Sunday, the four senators signaled they wanted to address the issue immediately in the organizing resolution for the new Senate.

"We look forward to working with you on this matter as the Senate works through the organizing resolution of the 115th Congress," they wrote.