And it wasn’t just voters who think Mr. King’s career is nearing an end: Republican officials are already planning to move past him, and are busy recruiting and raising money for primary challengers in 2020 if he does not step down.

Mr. King says he has no plans to seek “another line of work,” as Mitch McConnell of Kentucky, the Senate majority leader, recommended this week. He pushed back on Wednesday in a fund-raising letter to supporters, writing that “the unhinged Left has teamed up with Republican ‘NeverTrumpers’ and is pulling out all the stops to destroy me.”

He attacked The New York Times, which published a quotation from him in an article last week that led to the firestorm in Congress.

“White nationalist, white supremacist, Western civilization — how did that language become offensive?” Mr. King told The Times, comments he now says were taken out of context.

Some longtime supporters in Mr. King’s Fourth District said that even if the nine-term congressman defiantly stays on, they will most likely back a primary challenger. Two candidates announced this week that they would jump into the race, including a prominent Republican state lawmaker, Randy Feenstra, and others are expected.

The Fort Dodge Messenger, which in November endorsed Mr. King over his Democratic challenger, J.D. Scholten, issued an unusual correction on Wednesday night calling on Mr. King to resign.