Before then, though, investigators have scheduled almost a dozen additional private witness depositions, beginning Monday. It is unclear how many of those called to testify will appear, given Trump administration orders not to cooperate. Unlike earlier witnesses, many of whom were nonpartisan career officials, investigators are now trying to secure sessions with senior Trump administration political appointees who may be less likely to buck the White House’s wishes, even under subpoena.

Investigators issued subpoenas late Friday for John A. Eisenberg, the top legal adviser to the National Security Council, and Brian McCormack, the former chief of staff to Energy Secretary Rick Perry, to answer questions on Monday. It was unclear whether they would show up. A lawyer for at least one other witness scheduled for a Monday deposition, Robert Blair, indicated over the weekend that his client would not show up. Mr. Blair is the national security adviser to the acting White House chief of staff, Mick Mulvaney.

House Democrats have called Mr. Perry to sit for questioning on Wednesday, and John R. Bolton, the president’s former national security adviser who was said to be alarmed by pressure he saw being put on Ukraine, on Thursday. Their appearances are also in doubt.

Mr. McCarthy on Sunday also brought up another witness he wants to question: Mr. Schiff. He named the chairman of the Intelligence Committee and his aides as “the very first person we should bring.”

“He is the only person who knows who this whistle-blower is,” Mr. McCarthy said. “How many times did he meet with the whistle-blower? What did he talk to the whistle-blower about?”

The Republican leader was ostensibly referring to an October report by The New York Times that the C.I.A. officer approached a House Intelligence Committee aide with vague concerns about Mr. Trump, after he had a colleague first convey them to the C.I.A.’s top lawyer.

Allies of Mr. Trump on Capitol Hill have floated for weeks the idea of trying to question Mr. Schiff, but even some of the president’s staunchest defenders have brushed it off. Senator Lindsey Graham, Republican of South Carolina, told reporters last month that he was resisting pressure from Republicans to call Mr. Schiff to testify and that doing so would “cause a lot of damage to the country.”