“Now is not the time to process routine judicial nominations,” they added.

Mr. McConnell has been unapologetic about what his top priority is.

“As soon as we get back in session, we’ll start confirming judges again,” Mr. McConnell told the conservative radio host Hugh Hewitt last week. “We need to have hearings, and we need to confirm judges.”

Republicans have their eyes on confirming more than judges. The Senate Intelligence Committee could also convene a confirmation hearing next week for Representative John Ratcliffe, Republican of Texas, Mr. Trump’s nominee to be the director of national intelligence. Senators in both parties are eager to put in place a permanent spy chief, particularly at a time when intelligence gathering will play a large role in understanding the effects of the pandemic on American interests around the globe.

And Mr. Trump has agitated for the Senate to get back to work confirming his nominees, threatening this month that if it did not, he would use a never-before-invoked power to force Congress to adjourn so that he could fill government vacancies himself. (Mr. McConnell promptly issued a statement making it clear that would not happen, and saying that he had promised Mr. Trump he would work to confirm nominees who were “mission-critical to the Covid-19 pandemic.”)

Mr. Trump nominated Judge Walker, 37, to the prestigious appeals court on April 3, less than six months after he began serving on the district court bench. He was confirmed to that post last October despite being rated as unqualified by the American Bar Association because of his lack of experience. Mr. McConnell has known Judge Walker since he was in high school, considers him a brilliant legal mind and took him to the Oval Office this year to meet with the president.

A former clerk to Justice Brett M. Kavanaugh, Judge Walker was one of his most ardent defenders during the brutal Supreme Court confirmation battle in 2018. Mr. McConnell and Justice Kavanaugh traveled to Louisville in March for Judge Walker’s formal investiture as a district court judge before his latest nomination.

Progressive and civil rights groups opposed Judge Walker’s nomination initially and plan to do so again. In a letter sent to senators on Monday, the Leadership Conference on Civil and Human Rights urged a vote against his confirmation, calling his nomination “a triumph of nepotism over neutrality” given his close relationship with Mr. McConnell. The group faulted Judge Walker for his stated opposition to the Affordable Care Act and what it described as ideologically charged comments in defense of Justice Kavanaugh, among other concerns.