WASHINGTON — Moving quickly to force a showdown over voting rights, congressional Democrats plan to pursue a procedural shortcut to bring up stalled legislation for debate and try to win its approval over deep Republican resistance.

In a memo to Senate Democrats obtained by The New York Times, Senator Chuck Schumer of New York, the majority leader, on Wednesday laid out a new strategy intended to overcome at least one procedural obstacle erected by Republicans to prevent the Senate from even taking up the legislation.

Under the plan, the House would package two major pieces of voting rights legislation being pushed by Democrats — the Freedom to Vote Act and the John Lewis Voting Rights Advancement Act — insert them into an unrelated bill, and pass it. That would allow the bill to go to the Senate as what is known as a “message,” meaning a move to bring it up for debate in that chamber would not be subject to a filibuster, and Democrats would not need to muster 60 votes to proceed.

“Taking advantage of this existing exception to the Senate’s supermajority requirements will allow us to end the Republicans’ ability to block debate on voting rights legislation,” Mr. Schumer wrote in the memo. “The Senate will finally debate voting rights legislation, and then every senator will be faced with a choice of whether or not to pass the legislation to protect our democracy.”

Republicans have blocked four previous attempts to bring up voting rights legislation in the Senate, using the filibuster each time and leaving Democrats, who lack the 60 votes needed to get past the tactic, grasping for a way to even debate what they call vital measures to preserve democracy.

Even under the expedited process, the legislation could still face another filibuster when Democrats try to close off debate and bring it to a final vote, and Republicans have given every indication that they would try to block it.

If they do, Democrats have said they are ready to try to change Senate rules, a move endorsed by President Biden, who is scheduled to travel to Capitol Hill on Thursday to confer with Democrats on their strategy.

It was unclear how quickly the House would act, but Mr. Schumer has set Monday’s holiday for the Rev. Dr. Martin Luther King Jr. as a deadline.

The new approach came as Democrats stepped up their efforts to persuade their colleagues holding out against forcing through a rules change to join them, calling new voting restrictions being imposed at the state level a threat to democracy. Mr. Biden’s speech Tuesday in Atlanta, in which he compared opponents of the legislation to notorious Southern racists of the Jim Crow era, drew a stinging rebuke Wednesday from Senator Mitch McConnell, Republican of Kentucky and the minority leader.

“I’ve known, liked and personally respected Joe Biden for many years,” said Mr. McConnell, who said the president had reneged on his inaugural vow to try to unite country. “I did not recognize the man at the podium yesterday.”