(CNN) - A bipartisan deal to expand background checks on gun sales got a key conservative endorsement on Sunday, but one of its co-authors said Senate approval remained an "open question."
Sen. John McCain, R-Arizona, said he was "very favorably disposed" to the compromise measure that could come up for a vote as early as this week.
"Eighty percent of the American people want to see a better background check procedure," McCain said on CNN's "State of the Union," adding the country "wants to do what we can to prevent these tragedies and there's a lot more that needs to be done."
McCain's voice on the matter is important as a leading Senate conservative. He was one of the first to criticize a Republican-led filibuster around taking up gun-control legislation. The Senate overcame that opposition last week, voting to open debate.
Brokered by Sen. Joe Manchin, D-West Virginia, and Sen. Pat Toomey, R-Pennsylvania, the compromise proposal would extend background checks to include private purchases at gun shows and on the Internet.
Appearing on CNN with Manchin, Toomey told chief political correspondent Candy Crowley that he expected the measure to come up for a vote next week and it was an "open question" on whether it would pass.
Two other Republican senators, Susan Collins of Maine and Mark Kirk of Illinois, have come out in support of the measure. Sen. Marco Rubio, R-Florida, and others have been noncommittal, saying they want more time to read the proposal.
Sen. Dick Durbin of Illinois, who is responsible for lining up votes on legislation for majority Democrats, said on “Fox News Sunday” the compromise proposal had not been “whipped,” or counted, yet.
His counterpart on the Republican side, John Cornyn of Texas, would not say on the same program whether there were enough GOP members aligned to block the measure.
But there was negative Republican sentiment about the bill’s prospects on Sunday.
"I don't think it's going to pass," Sen. Jeff Sessions, R-Alabama, said on ABC's "This Week," arguing current gun laws are unenforced.
Another Republican, Sen. Mike Lee of Utah, also said he couldn't support the background checks compromise.
"This bill, I believe, would do more to limit the rights of the law abiding than it would to actually prevent violent crime, and that's why I can't support it," Lee said on NBC's "Meet the Press."
Some conservative Democrats are also on the fence, mindful of the pro-gun constituencies in the red states they represent.
"What we're asking for is just for our colleagues to read it," Manchin said, foreseeing greater support once skeptical lawmakers learn all of its components.
Himself a conservative Democrat representing a traditionally red state, Manchin said he wasn't calculating the political risk of pushing forward with gun control laws.
While he's had the backing of the National Rifle Association in the past, the group now forcefully opposes his initiative.
Manchin was clearly moved by the families of those killed in last December's school massacre in Connecticut, which jolted the country and prompted the current drive for stricter gun laws backed by President Barack Obama.
Manchin called them "the strongest people” he’d ever met with.
“They even said, ’We know that this bill that you're working on will not have saved our children. We know that. But it might save somebody else's child,’” Manchin recalled.
Toomey also claimed he wasn't overly concerned with the political blowback from conservative voters, saying he'd let "the political chips fall where they fall."
A background check compromise is expected to be unveiled in the House this week.