“He’s just a political hack. He’s trying to build a name for himself,” Mr. Trump said, adding, “It’s called presidential harassment. And it’s unfortunate. And it really does hurt our country.”

Mr. Schiff shot back, “I can understand why the idea of meaningful oversight terrifies the president. Look, several associates of his have gone to jail. Others are awaiting trial.”

He said the expanded Russia investigation would be done in coordination with other committees, including the Financial Services Committee, which wants to determine whether Russia or other foreign actors laundered money through the Trump Organization.

“Our job involves making sure the policy of the United States is being driven by the national interest,” Mr. Schiff said, “not by any financial entanglement, financial leverage or other form of compromise.”

Intelligence Committee Republicans, who have already begun to accuse the Democrats of politically motivated overreach, did not endorse the inquiry’s parameters. In a separate statement, they called on Democrats to subpoena unnamed witnesses they recommended, and privately, they accused Democrats of being more interested in keeping a politically damaging issue alive for the 2020 elections than finding new facts.

In its meeting, the committee also voted to share transcripts of witness interviews that it conducted related to Russian election interference with the special counsel, Robert S. Mueller III. Mr. Mueller has already used two such transcripts to charge associates of the president’s with lying to Congress, and Democrats believe others could have intentionally misled the committee.

Other committees were making moves, too.

The Judiciary Committee had called a meeting for Thursday to vote on a subpoena to compel testimony from Mr. Whitaker, if needed. Mr. Whitaker, a loyalist of the president’s who is currently overseeing Mr. Mueller’s work, is scheduled to testify voluntarily on Friday, but Democrats have concerns that he might try to back out or dodge questions about the firing of his predecessor, about the president’s attacks on the Justice Department and about other matters related to the Mueller inquiry.