WASHINGTON — Robert S. Mueller III, the special counsel, will not testify next week before Congress as Democratic lawmakers had hoped, though they are far from giving up on hearing from him despite President Trump’s broad objections to congressional investigations.

“It won’t be next week,” Representative Jerrold Nadler of New York, the chairman of the House Judiciary Committee, told reporters on Friday. “He will come at some point,” he added. “If necessary, we’ll subpoena him.”

Mr. Trump has waffled on whether House members could question Mr. Mueller about his investigation into Russian interference in the 2016 election and whether the president obstructed justice. On Thursday, Mr. Trump said he would leave the decision to Attorney General William P. Barr, who had previously said he had no problem with Mr. Mueller testifying on Capitol Hill.

Democrats view getting Mr. Mueller to testify before the Judiciary Committee as the most consequential of several fights Mr. Nadler is waging with the White House over access to the complete, unredacted special counsel report and its underlying evidence — including more details about the at least 10 instances of possible obstruction of justice that Mr. Mueller and his team investigated. Tensions have grown in recent days as the legislative and executive branches of government fight over constitutional powers.