Citing Republican obstruction, Democrats in 2013 changed Senate practices to end the 60-vote threshold for taking up judicial nominations, essentially allowing them to confirm judges on a simple majority vote. Earlier this year, Mr. McConnell led Republicans in taking that change one step further, eliminating the 60-vote threshold on Supreme Court nominees, culminating in the confirmation of Justice Neil M. Gorsuch.

But the blue slip practice — which is not a Senate rule but the prerogative of the Judiciary Committee chairman — remained in place. Because Democrats abided by it during Barack Obama’s presidency and Republicans refused to sign off on many of his nominees, his tenure ended with a large number of court openings that the Trump White House can now fill if it can overcome Democratic objections.

At the moment, there are 144 federal court vacancies, including 21 on the appeals courts and 115 at the district court level. Besides Justice Gorsuch, Mr. Trump has won the confirmation of three appeals court judges and two district judges. Forty-five nominations are pending — 11 for the appeals courts, and 34 at the district level.

A few of those nominations have already run into what is known in the Senate as blue slip trouble, including that of David R. Stras, a conservative favorite from Minnesota nominated by Mr. Trump in May to the United States Court of Appeals for the Eighth Circuit.

Senator Al Franken, Democrat of Minnesota and a member of the Judiciary Committee, earlier this month announced that he would not return his blue slip for Mr. Stras, citing a fear that he was too much in the mold of Justices Clarence Thomas and Antonin Scalia.

At the same time, Senators Ron Wyden and Jeff Merkley of Oregon, both Democrats, announced that they would not return a blue slip for Ryan Bounds, newly nominated to a seat on the Court of Appeals for the Ninth Circuit, because the White House had failed to engage in an existing judicial recommendation process in the state. Under the existing system, the decisions by the senators could permanently derail those nominees.

Equally worrisome to Republicans is the fact that many of the other appeals court openings are in states represented by either one or two Democrats, potentially giving them significant sway over multiple nominees despite their minority status.