In recent days, some of the most prominent figures in the G.O.P. outside the White House have broken with Mr. Trump over issues like the value of wearing a mask in public and heeding the advice of health experts like Dr. Anthony S. Fauci, whom the president and other hard-right figures within the administration have subjected to caustic personal criticism.

They appear to be spurred by several overlapping forces, including deteriorating conditions in their own states, Mr. Trump’s seeming indifference to the problem and the approach of a presidential election in which Mr. Trump is badly lagging his Democratic challenger, Joseph R. Biden Jr., in the polls.

Once-reticent Republican governors are now issuing orders on mask-wearing and business restrictions that run counter to Mr. Trump’s demands. Some of those governors have been holding late-night phone calls among themselves to trade ideas and grievances; they have sought out partners in the administration other than the president, including Vice President Mike Pence, who, despite echoing Mr. Trump in public, is seen by governors as far more attentive to the continuing disaster.

“The president got bored with it,” David Carney, an adviser to the Texas governor, Greg Abbott, a Republican, said of the pandemic. He noted that Mr. Abbott directs his requests to Mr. Pence, with whom he speaks two to three times a week.