A tea-party hero and 2016 supporter of President Trump is expected to announce that he will challenge the president for the 2020 Republican nomination, The New York Times reported Wednesday evening.

Former Rep. Joe Walsh, Illinois Republican and now a talk-radio host, has become more critical of Mr. Trump (he recently called him a “racial arsonist”) and wants to appeal to conservative looking for a Trump alternative.

Citing “two people who have spoken to him,” The Times reported that he has “hired a senior political adviser, organized meetings with high-profile Trump antagonists in New York City, and published an Op-Ed in The New York Times with the goal of previewing his campaign message.”

Mr. Walsh, who served one term in the U.S. House, was coy to The Times when asked directly about running, answering in the conditional tense.

“If I do it, it’s going to be before Labor Day,” he said.

No sitting president who wanted his party’s nomination for a second term has failed to get it in the modern primary system, though Lyndon Johnson dropped out in 1968 rather than risk losing.

Team Trump was not worried about Mr. Walsh.

Campaign spokesman Tim Murtaugh, when asked by The Times for comment, replied with two words: “Certain failure.”

Mr. Trump already has one official challenger, former Massachusetts Gov. William Weld, but he is running against Mr. Trump from his left, not the path Mr. Walsh is likely to take, one prominent Republican Trump critic told The Times.

“He has a different appeal than Bill Weld,” William Kristol said. “The fact that he was a Tea Party congressman who voted for Trump in 2016 gives him an ability to speak to Republican primary voters that ‘Never Trumpers’ like me don’t have.”

Mr. Kristol, the former Weekly Standard editor and vocal Trump critic, has been trying to recruit a Trump challenger for the 2020 primaries for more than a year, The Times reported.

The president was asked earlier Wednesday what he thought about Republicans who might challenge him in the GOP primary.

“I’m at 94 percent now in the Republican Party — the highest in history, the highest of any Republican,” the president told reporters. “So, I think they’d have a hard time.”

• Dave Boyer contributed to this report.

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