Mr. Schumer and Democrats called on their former Senate colleague, Jeff Sessions, the new attorney general and a top Trump campaign supporter, to recuse himself from any review. (Advisers to Mr. Sessions have said he sees no need to do so.) They demanded that any relevant records from the administration, transition or campaign be preserved. And they asked that former campaign and transition associates testify “in public, under oath,” a request Republicans have not echoed.

Pressed on how he hoped to proceed if Republicans declined to join his calls, Mr. Schumer demurred. “Let me just say, it’s the right thing to do,” he said. Throughout his news conference, the tone seemed telling, signaling Democrats’ reliance on at least some Republican support for their efforts: Absent were any taunting attacks on Republicans — as have been common in the fight over Mr. Trump’s cabinet selections — replaced by earnest-sounding pleas for collaboration.

At the same time, other Democrats seethed over what they saw as a double standard at the F.B.I. before the election, when the Trump campaign’s possible ties to Russia first received attention.

Last summer, James B. Comey, the F.B.I. director, broke with protocol and tradition when he held a news conference to discuss the agency’s decision not to bring criminal charges against Hillary Clinton for keeping classified information on a private email server. Then, days before the election, he informed Congress — and by extension the public — that the F.B.I. was investigating newly discovered emails related to that case.