On a Saturday in early March, Donald J. Trump, clad in a baseball cap, strode into the Situation Room for a meeting with the coronavirus task force. He didn’t stop by the group’s daily meetings often, but he had an idea he was eager to share: He wanted to start a White House talk radio show…

But that Saturday, almost as suddenly as he proposed it, the president outlined one reason he would not be moving forward with it: He did not want to compete with Rush Limbaugh.

No one in the room was sure how to respond, two of the officials said. Someone suggested hosting the show in the mornings or on weekends, to steer clear of the conservative radio host’s schedule. But Mr. Trump shook his head, saying he envisioned his show as two hours a day, every day. And were it not for Mr. Limbaugh, and the risk of encroaching on his territory, he reiterated, he would do it.