Representative Jerrold Nadler of New York, the chairman of the House Judiciary Committee, said Mr. Mueller was the only person who could clear up certain ambiguities about the report. Only Mr. Mueller, he said, could tell the American people “if he agrees with the fact that if he were not president, he would have been indicted” for the instances of obstruction identified in the report.

But talking with reporters at the Capitol on Thursday, Mr. Nadler conceded that the White House strategy had thus far succeeded in tamping down energy around the Mueller report and investigations. However, he added, “the temperature can rise very quickly when the first subpoena is adjudged in our favor, and we start getting witnesses.”

Over the past week, aides with the House Judiciary Committee have been negotiating with aides to Mr. Mueller to get the special counsel, who remains an employee of the Justice Department, to testify. Those talks grind on over the format and the length of his appearance, according to two people close to the deliberations.

It is not clear if he would appear alone or with key aides who helped draft the report, they said. Some committee Democrats have expressed the opinion that two top Mueller aides, Aaron M. Zebley and Andrew Weissmann, would feel less constrained about criticizing the president than Mr. Mueller.

Mr. Schiff’s staff has also been talking to Mr. Mueller’s office, and the congressman expressed optimism that a deal could be struck. “I think we’ll get there,” he said.

Peter Carr, a spokesman for the special counsel, declined to comment.

In the past, there has been jockeying between Mr. Nadler’s staff and aides to Mr. Schiff — but they are in agreement on sequencing. If Mr. Mueller agrees to appear, he would testify in an open session before the Judiciary Committee first, then appear before Mr. Schiff’s committee, most likely in public and closed-door sessions, Mr. Nadler said.

Mr. Trump has given conflicting answers over his feelings about Mr. Mueller’s testimony, after labeling his investigation a “witch hunt” in the months leading up to investigators’ conclusion that there was insufficient evidence to charge Mr. Trump, his campaign or his supporters with conspiring with the Kremlin.