Instead, the Republican leader is taking a more measured approach, as GOP senators are talking frequently among themselves, and with the White House, in the face of mounting criticism that Congress is failing to act.
President Donald Trump is privately calling up senators — and publicly pushing for an expansion of background checks for firearms purchases — but McConnell knows those ideas have little Republican support. In fact, the White House threatened to veto a House-passed background checks bill earlier this year. Yet, as the nation reels from the frequency of shootings and their grave toll, McConnell's unwillingness to confront the gun lobby or move more swiftly is coming under scrutiny.
"I can only do what I can do," the president told reporters as he departed Washington for visits to Dayton, Ohio, and El Paso, Texas, where 31 people were killed in two mass shootings over the weekend.
On Wednesday, Ohio Democratic Sen. Sherrod Brown made a personal plea to Trump during his visit to "call on Sen. McConnell to bring the Senate back in session this week, to tell the Senate he wants the background checks bill that has already passed the House."
House Democrats signed onto a letter urging McConnell to act immediately on the House-passed legislation, which would require federal background checks for all firearms sales and transfers, including online and at gun shows. In Kentucky, where McConnell is recuperating from a weekend fall that left his shoulder fractured, activists have been demonstrating at his home and protesting at his downtown Louisville office.
"House Democrats are moving prayerfully and purposefully to advance action," wrote Speaker Nancy Pelosi in a letter Wednesday to Democratic colleagues. The Judiciary Committee, led by Rep. Jerry Nadler, D-N.Y., may take action during the recess on so-called "red flag" legislation to allow removal of guns from those deemed a threat to themselves or others.
But none of it has moved the Republican Senate to act more urgently. McConnell's office is declining comment, referring back to a short statement he issued late Monday saying he was tasking three GOP committee chairmen "to engage in bipartisan discussions of potential solutions."
The politics of gun violence are difficult for Republicans, including McConnell, who would risk losing support as he seeks reelection in Kentucky if he backed restricting access to firearms and ammunition. Other Republicans, including those in Colorado, Maine and swing states, also would face difficult votes, despite the clamor for some changes to gun laws.
"In Congress, we're trying to come up with some answers," Texas GOP Sen. John Cornyn, who is also up for reelection, said after donating blood in El Paso. In the meantime, Trump has been dialing up Senate Republicans about what is possible. Trump spoke at least three times with Sen. Pat Toomey, R-Pa., who has long pushed a bipartisan background check bill with Sen. Joe Manchin, D-W.Va., the GOP senator said.
Trump continues to say there's "great appetite" for background checks legislation. "I think we can bring up background checks like we've never had before," he said before departing Washington. But that is not the case, for now.
Instead, Republicans are trying to build support for more modest measures, including so-called red-flag bills from Sen. Lindsey Graham, R-S.C., and Sen. Marco Rubio, R-Fla., that would allow friends and family to petition authorities to keep guns away from some people. But those efforts are also running into trouble from conservatives, who worry about due process and infringing on gun owners' rights.
GOP senators are also considering changes to the existing federal background checks system, modeled on the so-called "fix-NICS" bill signed into law from the last session of Congress, as well as strengthening penalties for hate crimes, Republicans said.
Republican senators have been doing almost daily conference calls, and talking among themselves and the White House, as they try to figure out their next steps, according to a Republican aide familiar with calls who discussed the private talks on condition of anonymity.
While many of those proposals have bipartisan support, Democrats are unlikely to agree to them without consideration of the more substantive background checks bill. Senate Minority Leader Chuck Schumer said Wednesday, "We Democrats are not going to settle for half-measures so Republicans can feel better and try to push the issue of gun violence off to the side."
Even though the Democratic-led House overwhelmingly approved the background checks bill in February, it has scant support in the Republican Senate. The version from Toomey-Manchin had its greatest backing during a 2013 vote, in the aftermath of the Sandy Hook school shooting, but it failed to reach the 60-vote threshold needed to advance.
Since then, Toomey and Maine Sen. Susan Collins are the only two Republicans who voted for the background checks bill in 2015 — the last time it was brought to the floor in the Senate. But Toomey said he thinks sentiment in the Senate has moved toward the bill in recent years and he was heartened by Trump's "encouraging remarks," he tweeted Wednesday.
At least one Republican, Rep. Mike Turner of Ohio, who had voted against the House-passed bill in February, said in a statement after the shooting that he supports other gun limits, including banning the sale of assault weapons to civilians, limiting the size of magazines and enacting red-flag laws.
Turner, whose district includes Dayton, said the "carnage these military-style weapons are able to produce when available to the wrong people is intolerable." Turner's daughter and a family friend were at a bar across the street from where the shooting began.
Associated Press writer Bruce Schreiner reported from Louisville.