Speaker Mike Johnson on Thursday wrote to Senator Chuck Schumer, the majority leader, demanding that the Senate hold an impeachment trial next month of Alejandro N. Mayorkas, the homeland security secretary. The speaker called plans to quickly dismiss the charges against Mr. Mayorkas “a violation of our constitutional order and an affront to the American people.”

In a letter signed by the 11 Republicans he named as impeachment managers to try Mr. Mayorkas, Mr. Johnson wrote that they planned to send over the charges on April 10, after the Senate returns from its Easter break. Senators in both parties, who serve as the jury for impeachment trials, have indicated that they do not want to sit through such a proceeding in the case of Mr. Mayorkas, notwithstanding House Republicans’ insistence on it.

The letter rehashed the accusations against the homeland security secretary, with signatories including Representatives Mark E. Green of Tennessee, the chairman of the Homeland Security Committee, and Representative Marjorie Taylor Greene of Georgia, who introduced articles of impeachment against Mr. Mayorkas. The Republicans condemned what they called a “willful and systemic refusal to comply with the law and his breach of the public trust,” accused the secretary of lying to Congress, and blamed him for the state of the southern border with Mexico.

There is little doubt that the Senate, which is controlled by Democrats, will side with Mr. Mayorkas. Leaders are expected to dispense with a trial quickly, either by dismissing the charges immediately or moving to a quick vote in which Republicans have no chance of securing the two-thirds necessary to convict and remove Mr. Mayorkas.

Mr. Schumer has called the impeachment effort a “sham” and “another embarrassment for House Republicans.”

“House Republicans failed to produce any evidence that Secretary Mayorkas has committed any crime,” he said in a statement Thursday. “House Republicans failed to show he has violated the Constitution. House Republicans failed to present any evidence of anything resembling an impeachable offense. This is a new low for House Republicans.”

House Republicans impeached Mr. Mayorkas in February by a single vote, racing ahead with a case that constitutional scholars called groundless before Democrats won a special election on Long Island and wiped away the majority support to approve the charges. But instead of quickly sending the articles over to the Senate to try to force out one of the officials they blame for chaos at America’s southern border, Republicans have sat on them.

G.O.P. lawmakers are hoping to build public pressure for a full trial, which would generate media coverage of their accusations against President Biden’s top immigration official. That could give them a high-profile platform for one of their biggest election-year attacks on the president and Democrats — a clear political boon even if Mr. Mayorkas is ultimately acquitted.

“We call upon you to fulfill your constitutional obligation to hold this trial,” the letter stated. “The American people demand a secure border, an end to this crisis, and accountability for those responsible. To table articles of impeachment without ever hearing a single argument or reviewing a piece of evidence would be a violation of our constitutional order and an affront to the American people whom we all serve.”

While some Senate Republicans, such as Senator Mike Lee of Utah, have demanded a trial, others have been less enthusiastic about the impeachment effort, arguing the floor time would be better used on less partisan matters.

Mia Ehrenberg, a spokeswoman for the Department of Homeland Security, said the Republicans’ impeachment effort “will be remembered by history for trampling on the Constitution for political gain rather than working to solve the serious challenges at our border.”

“Without a shred of evidence or legitimate constitutional grounds, and despite bipartisan opposition, House Republicans have falsely smeared a dedicated public servant who has spent more than 20 years enforcing our laws and serving our country,” Ms. Ehrenberg said.