The White House declined to comment on Mr. Trump’s desire for a radio show. The president ultimately opted for daily televised press briefings instead, which have in effect served as a stand-in for campaign rallies and regularly span two hours.

“The Rush Limbaugh Show” has been the most popular talk-radio show in the country for decades, currently drawing 15.5 million listeners a week. In that time Mr. Limbaugh has traded in the kind of deeply divisive messaging that Mr. Trump regularly brandishes to appeal to his conservative base.

Like the president, Mr. Limbaugh has also dispensed disinformation and falsehoods at a rapid clip. In the last few weeks alone he has repeatedly referred to the coronavirus as the “common cold.” His history of anti-gay remarks was revived as recently as February, when he said Americans would not elect Pete Buttigieg after seeing him “kissing his husband onstage, next to Mr. Man, Donald Trump.” (Mr. Limbaugh later told listeners that Mr. Trump had called him and told him not to apologize for the comments.)

Mr. Limbaugh is among Mr. Trump’s most influential backers, praising him for his politics and leadership well before many other Republicans cast their lot with him. He recently called attacks on the president’s handling of the virus crisis “a political hit job.” The president has returned the favor: In a surprise move during his State of the Union address in February, he awarded Mr. Limbaugh, who had recently revealed he had late-stage lung cancer, the Presidential Medal of Freedom, the nation’s highest civilian honor. He highlighted Mr. Limbaugh’s charitable work and called him “the greatest fighter and winner that you will ever meet.”

As public health officials urge caution in relaxing stay-at-home restrictions, corporate executives and conservative activists alike are pleading with the president to reopen the economy. Mr. Trump has argued that he alone has the power to override stay-at-home orders imposed at the state level — a claim that legal scholars reject and that he appeared to walk back on Tuesday. Still, many Republican governors are looking directly to the White House to set the tone for a path forward.

As for his own guidance, Mr. Trump’s preferred media sources have tempered their advocacy. After initially dismissing the severity of the virus, many of Mr. Trump’s go-to talkers on Fox News have approached the question with relative delicacy, appearing to take their cues from the president rather than try to proactively urge one path or another.

Mr. Limbaugh, conversely, has been a forceful voice from the get-go. Along with casting medical experts like Dr. Anthony S. Fauci as “Hillary Clinton sympathizers,” he has used his enormous platform to call for a rapid return to normal life.