“I have urged all parties to come together and come up with a responsible gun safety package that can pass the Senate,” said Senator Susan Collins, Republican of Maine.

Mr. McConnell is famously loath to press forward with any legislation that sparks division among Republicans. He had to be strong-armed into bringing a criminal justice overhaul to the floor despite backing from the president and the vast majority of Senate Republicans because of the objections of a handful of vocal conservatives.

He also often cites an unwillingness to press legislation that Mr. Trump does not wholly support, saying it is a waste of time to send legislation to the president’s desk only to draw a veto. “I want to make a law, not just see this kind of political sparring going on endlessly, which never produces a result,” he said during an interview with a Louisville radio station when discussing gun legislation this month.

So if Mr. McConnell is looking for a way out of taking up gun safety legislation, he can just point to the president’s perceived opposition, throw in some criticism of Democrats and gun control activists for overreaching and try to move on.

Democrats say they do not intend to make it easy for him to do so.

“Senator McConnell has been begging President Trump to let him off the hook when it comes to passing universal background checks legislation to address the gun violence epidemic,” said Senator Chuck Schumer of New York, the Democratic leader. “But on behalf of the families of the victims of gun violence and the 90 percent of Americans who support universal background checks, Senate Democrats will keep the pressure up and not let Senator McConnell off the hook.”

A refusal to take action not only would draw the condemnation of Democrats, but it could also further endanger some Republicans facing re-election — and, by extension, the party’s control of the Senate. Despite strong opposition from the gun lobby, expanded background checks and some other gun restrictions draw support from most voters in polls, particularly the women and suburbanites that senators such as Cory Gardner, the embattled Colorado Republican, will need next year to hold on to their seats.

Gun safety advocates say that they employed the issue in 2018 to oust a House Republican from Colorado, and that they intend to keep the heat on Mr. McConnell.