“I literally cannot walk across the parking lot without being stopped to talk about this issue,” he said. “I haven’t seen anything quite like this.”

He is hardly alone. Fewer than a dozen House Republicans, a total that includes the top two leaders, have publicly said they would back the president on a military strike, making the White House climb to a House majority exceedingly steep given significant Democratic resistance as well. Not only is the administration not winning over Republicans, it lost at least one it had. Representative Michael G. Grimm, Republican of New York, said Thursday that he was reversing his support. “The moment to show our strength has passed,” he said.

Mr. Cole’s constituent experience is not isolated. Representative Mick Mulvaney, a Republican swept into power in 2010 in military-focused South Carolina on a platform of small government, said that in his three-plus years in Congress, no issue had elicited as passionate a response as Syria. And, he added, “to say it’s 99 percent against would be overstating the support.”

Of the 1,000 or so calls and e-mails he has received, three supported some kind of response. And two-thirds of the correspondents have never reached out to him before.