Speaker Nancy Pelosi and other Democratic leaders have conspicuously avoided locking in a timeline for the inquiry, but privately they are said to be aiming for a full House vote on impeachment articles before the Christmas recess, barring unexpected developments. That would leave the Judiciary Committee with as little as two weeks to do its work.

“Even at this late date,” Mr. Cipollone wrote, “it is not yet clear whether you will afford the president at least these basic, fundamental rights, or continue to deny them.”

Mr. Nadler, in consultation with Ms. Pelosi and his members, will have to decide how to handle requests like Mr. Cipollone’s, weighing a desire to demonstrate fairness to Mr. Trump against a determination to maintain forward momentum in the proceedings. It is one of the many delicate tasks, fraught with political risks and legal intricacies, that have fallen to the judiciary panel as the impeachment inquiry enters a critical phase.

The first milestone will come in the form of a written report from the Intelligence Committee, which is to be approved on Tuesday. The handoff of the report, which will most likely form much of the basis for articles of impeachment against Mr. Trump, will be a stylistic and substantive turning point for the inquiry that will almost certainly inflame a debate that has already roiled Congress and divided the country.

Large, disorderly and stacked with some of Congress’s most outspoken progressives and conservatives, the Judiciary Committee is the polar opposite of the small and staid intelligence panel, where rules drafted to facilitate the handling of government secrets allowed Democrats to tightly control every aspect of the impeachment inquiry.

The judiciary rules, instead, are fundamentally democratic, intended to provide wide latitude for divisive debates over the nation’s most pressing policy issues, many of the them cultural hot buttons that fuel each party’s activist base. Barring some momentous new evidence, not a single lawmaker on either side is expected to budge.

And while the Intelligence Committee conducted much of its investigative work behind closed doors, the judiciary panel will work entirely in the public glare.