One top Republican aide said that unless the president gave lawmakers cover on background checks, it was not clear what could be accomplished.

Democrats, who have watched Mr. Trump play to his base in recent months, with his frequent attacks on four Democratic congresswomen of color, as well as on Representative Elijah E. Cummings, Democrat of Baltimore, said spearheading any significant action on gun legislation now would seem to run counter to his re-election playbook.

Some gun control advocates hoped the calculus would be different for Republicans on Capitol Hill.

“If I were a Republican senator up in 2020, I’d be asking myself three things,” said John Feinblatt, the president of Everytown for Gun Safety, a national group that advocates tougher gun laws. “How many women are in my state who expect me to be voting on gun safety, how many young people are expecting me to do something to protect them, and how bad the dumpster fire is over at N.R.A. headquarters.”

Mr. Feinblatt also noted that he had never seen the N.R.A., which is under investigation by attorneys general in New York and Washington, D.C., and mired in legal battles and internal riffs, seem “weaker.” He said Republicans were “looking at the polls, and certainly looking at the suburbs and balancing that against whether the N.R.A. has any muscle left.”

In private, Mr. Trump has echoed Mr. Feinblatt’s bleak assessment of the N.R.A. But aides have warned him that its members are among his voters, and they may be less attuned to the internal drama at the organization than to its mission statement.

Officials at the gun rights group have been looking to show it still has muscle amid a series of stories about its finances and management, and responding to Mr. Trump has given them an opportunity to do that.

Officials at the organization would not directly comment on discussions with the president or his staff. But Andrew Arulanandam, a spokesman for the group, said, “If anyone looks at it from a very logical standpoint, they’ll realize that the sound bite remedy being offered by many just won’t work.”