On the same day former President Donald J. Trump vowed that if elected he would immediately fire Jack Smith, his lawyers sought to move ahead with an effort to have the federal election interference case against Mr. Trump dismissed on the grounds that Mr. Smith had been illegally appointed to his job as special counsel.

The request to launch a fresh attack on the indictment brought by Mr. Smith charging Mr. Trump with seeking to overturn the results of the 2020 election came as the 2024 election sped toward its conclusion.

“The proposed motion establishes that this unjust case was dead on arrival — unconstitutional even before its inception,” Mr. Trump’s lawyers wrote in papers filed in Federal District Court in Washington on Thursday.

The lawyers went on to say that they believe Attorney General Merrick B. Garland violated the Appointments Clause of the Constitution “by naming private citizen Smith to target President Trump, while President Trump was campaigning to take back the Oval Office from the attorney general’s boss, without a statutory basis for doing so.”

Mr. Trump’s lawyers were required to ask Judge Tanya S. Chutkan, who is overseeing the election case, for permission to file their motion in part because a federal appeals court in Washington had already weighed in on the question, ruling that the appointment of special counsels like Mr. Smith is in fact valid.

Both Judge Chutkan and the appeals court are unlikely to look with much favor on Mr. Trump’s attempt to challenge Mr. Smith’s appointment.