WASHINGTON — Republican lawmakers warned President Trump on Sunday not to fire Robert S. Mueller III, but showed little sense of urgency to advance long-stalled legislation to protect the special counsel despite a report that Mr. Trump had tried to remove him last June.

“I don’t think there’s a need for legislation right now to protect Mueller,” Representative Kevin McCarthy of California, the House majority leader, said Sunday on NBC’s “Meet the Press.” “Right now there’s not an issue. So why create one when there isn’t a place for it?”

Mr. McCarthy’s comments, similar to those made earlier by other Republicans, come amid bipartisan outrage over a report last week in The New York Times that Mr. Trump sought in June to fire Mr. Mueller, who is investigating Russian interference in the 2016 election.

The president backed down only after Donald F. McGahn II, the White House counsel, threatened to quit rather than execute Mr. Trump’s order.