WASHINGTON — Senator Mitch McConnell says he is awaiting President Trump’s proposal on new gun safety rules before swinging his chamber into action. Some of his colleagues think he’s got it backward.

After all, it is the Senate — in conjunction with the faster-moving House — that by tradition has jealously guarded its role in originating and shaping legislation, only later sending it to the president to be accepted or rejected. But today’s Senate, devoted almost exclusively to confirming Mr. Trump’s nominations, is hardly a hotbed of legislative activity.

Members of both parties say they would like that to change.

“I’m very eager to turn from nominations to legislation,” said Senator Susan Collins, Republican of Maine. “There are important issues that are pending, and I think we could produce some terrific bills that would be signed into law.”

Democrats took a harsher line, particularly when it comes to stricter gun safety legislation they are demanding after an August marked by mass shooting sprees.