While President Donald Trump has said he would veto a House-passed bill to expand background checks for gun purchases, McConnell said he is hopeful there are other gun-related proposals that Congress can approve and Trump can support.
"I still await guidance from the White House as to what (Trump) thinks he's comfortable signing," the Kentucky Republican told reporters. "If and when that happens, then we'll have a real possibility of actually changing the law and hopefully making some progress."
Senate Democratic Leader Chuck Schumer said McConnell and Trump were blocking meaningful action on gun violence, adding, "This is the moment for the president to do something different and courageous."
The New York Democrat said he wonders whether Trump will "rise to the occasion, or will he squander this opportunity as he always has done in the past?"
Schumer and House Speaker Nancy Pelosi warned Trump on Sunday that any proposal on gun control must include the House-passed bill to expand background checks. Pelosi and Schumer spoke with Trump by phone and said they made it clear any proposal that does not include the House legislation "will not get the job done" because dangerous loopholes will be left open.
Schumer said Tuesday he was "not encouraged by what the president said," but remained committed to pushing for stricter gun control measures. Senate Democrats planned to speak for hours on the Senate floor Tuesday to urge passage of background checks and other measures in the wake of mass shootings in Texas and Ohio last month that killed more than three dozen people.
Trump and White House aides have discussed a number of gun-control measures with members of Congress, including steps to go after fraudulent buyers, notify state and local law enforcement when a potential buyer fails a background check, issue state-level emergency risk protection orders, boost mental health assistance and speed up executions for those found guilty of committing mass shootings.
Trump hopes to reveal something on gun control to the American public "very soon," White House spokesman Hogan Gidley said Tuesday. The White House expects the gun proposal later this week or early next week, according to a person familiar with the administration's thinking.
Attorney General William Barr and White House legislative affairs director Eric Ueland met with GOP senators Tuesday to talk about a path forward. Senate Judiciary Committee Chairman Lindsey Graham, R-S.C., said background checks remained under discussion, but it was not clear whether progress was being made.
Sen. John Cornyn, R-Texas, said background checks did not come up during a lunch meeting Tuesday between Senate Republicans and Vice President Mike Pence.
Sen. John Thune, R-S.D., cautioned against overinterpreting the relative silence by the White House. "My guess is they're still vetting ideas, proposals and kind of putting together their plan," he said.
Sen. Joe Manchin, a West Virginia Democrat who has helped lead a bipartisan push to expand background checks, said he had not spoken to Trump since late last week. Manchin said he considers a proposal he is offering with Pennsylvania Republican Sen. Pat Toomey a starting point for legislative action.
"You can't water it down because that's the bedrock," Manchin said, adding that senators and the White House haven't agreed on anything yet. "We're just going to see where it goes," he said.