The aftermath of the back-to-back massacres in El Paso and Dayton that killed 31 people has created an opening for Mr. Trump, who has been encouraged by advisers that he could accomplish something that his predecessors were unable to, and who has told reporters that he wants to accomplish “meaningful background checks.” And the president has assured advisers that he thinks the powerful National Rifle Association, which is rife with internal conflict, is “going bankrupt.”

But Mr. Trump is still facing pushback from gun rights advocates and some of his own advisers, and in the end, the president could repeat his pattern on other issues of abruptly dropping the idea of doing something after talking about it for weeks.

“He has to do some of the Lyndon Johnson ‘come, let us reason together,’ real down-to-earth stuff that Johnson did,” said Senator Richard Blumenthal, a Connecticut Democrat who is a leading proponent of gun safety, invoking the former president and onetime Senate majority leader who was famous for leaning on lawmakers — not only figuratively but sometimes physically — to get the legislation he wanted.

“If Donald Trump is standing there,” Mr. Manchin said, “and we’ve got bipartisan senators standing beside him, and he says, ‘This is my piece of legislation and I support it wholeheartedly,’ it’ll pass. Without that, we’re back to square one.”

A White House spokesman did not respond to a request for comment.

But Mr. Trump told advisers shortly after the shootings less than two weeks ago that he wanted to put in place some type of measure restricting who could buy guns. And aides privately anticipate that he will try to convince senators who are uncertain about backing legislation related to guns once the Senate returns to Washington.