Even before Thursday, the Intelligence Committee inquiry had been viewed skeptically.

Last week, The Washington Post reported that Mr. Burr spoke with the White House about Russia-related news reports and engaged with news organizations to dispute reports that associates of Mr. Trump had consistent contact with Russian intelligence operatives during the campaign.

Senator Mark Warner of Virginia, the ranking Democrat on the committee, expressed “grave concerns” about Mr. Burr’s contact with the White House. But despite Democrats’ deep reservations about Mr. Burr’s capacity to carry out a comprehensive investigation, many party leaders still view the committee’s work as the best hope of holding Mr. Trump and his associates to account.

On Thursday, Mr. Schumer said that if the Justice Department would not appoint an independent special prosecutor, Congress should intervene by reviving an independent counsel law put into place after Watergate. He also asked the department’s inspector general to begin an immediate inquiry into Mr. Sessions “to discover if the investigation has already been compromised.”

Mr. Schumer sought to distinguish between Mr. Sessions’s meeting with the Russian ambassador, Sergey I. Kislyak, which he called appropriate, and the “very inappropriate” step of misleading Congress. (One Democrat, Senator Claire McCaskill of Missouri, invited criticism on Thursday for saying that she had never held a call or meeting with the Russian ambassador, despite previously describing two such conversations on Twitter. She blamed Twitter’s character limit for the confusion.)

At his confirmation hearing, Mr. Sessions said he “did not have communications with the Russians” during the presidential campaign. But the Justice Department now acknowledges that Mr. Sessions twice communicated with Mr. Kislyak last year: once after a speech at the Republican National Convention and once in his office.