Trump associates who believe they were unfairly maligned in Robert Mueller’s Russia report want lawmakers to ask one question — why? — when the former special counsel testifies July 17.

They also allege that Mr. Mueller knew there was no Trump-Russia conspiracy months before the March 22 completion date of his investigation but stayed quiet. Why? They say the report contains pejorative narratives of innocent conduct for which no one was criminally charged.

Michael Caputo, media adviser for the Trump campaign, asked on his “Still Standing” podcast, “When did Bob Mueller actually know there was no Russian collusion?

“That matters to me. That matters a lot,” Mr. Caputo said, adding that if Mr. Mueller knew before the 2018 midterm elections, “he was rather duty-bound to announce that.”

Republicans lost their House majority at a time when Democrats and the news media accused Mr. Trump of being a Russian agent and involved in a conspiracy with the Kremlin.

“Did Mueller sit on his ‘no collusion’ decision?” Mr. Caputo asked. “What did Robert Mueller know, and when did he know it?”

George Papadopoulos, one of the most FBI-scrutinized Trump campaign volunteers, told The Washington Times that he has a list of questions for Mr. Mueller.

U.S. intelligence assigned at least two spies — college professor Stefan Halper and a supposed associate, Azra Turk — to investigate whether Papadopoulos would admit to colluding with Russia. Papadopoulos was in London trying to set up a Kremlin-Trump meeting in 2016.

Papadopoulos always has maintained his innocence. He was never charged in such a conspiracy, but he pleaded guilty to lying to the FBI.

“Were Stefan Halper and Azra Turk working with the CIA?” Papadopoulos asked.

“Who was Charles Tawil working for?” he asked, referring to an American-Israeli businessman who handed him $10,000 in cash in a hotel room in Tel Aviv as a consultant’s retainer.

“What kind of businessman gives someone $10,000 — in cash — without a receipt or a contract?” Papadopoulos wrote in his book, “Deep State Target: How I Got Caught in the Crosshairs of the Plot to Bring Down President Trump.” “A terrible one, someone used to paying bribes, or a businessman who is not really a businessman.”

On returning to the U.S. via Washington Dulles International Airport, Papadopoulos was met by FBI agents who vigorously searched his luggage for the Tawil cash, he said. At one point, the FBI threatened to charge him with being an agent of Israel. Under that scenario, the $10,000 would have been evidence of such an arrangement.

Mr. Tawil has denied he is a Western agent. He said Papadopoulos asked for the money as a down payment for energy consulting.

“Who was Joseph Mifsud working for?” Papadopoulos asked, referring to a Maltese professor who ingratiated himself to the Trump volunteer in London.

Mr. Mifsud told Papadopoulos in April 2016 that he had heard Russia had Hillary Clinton emails. The conversation ultimately persuaded the FBI to open an investigation.

Trump associates say that as a general policy, Justice Department prosecutors don’t talk about unindicted people once a case is closed.

Former New Jersey Gov. Chris Christie, a former federal prosecutor, told ABC News as the Mueller report was imminent: “If you charge, you can say whatever you are willing to do an indictment and have a grand jury sign off on; if you don’t charge, keep quiet. You’re not supposed to talk about people you’re unwilling to charge.”

But that is exactly what Mr. Mueller did.

In his 448-page report, the special counsel made an overriding conclusion: His large prosecutor/FBI/intelligence team failed to establish a conspiracy between the Trump campaign and Moscow to interfere in the 2016 election by computer hacking and trolling. Not one was charged in a conspiracy.

Yet the names of about three dozen Trump associates appeared multiple times, often in negative ways. Their conduct wasn’t criminal. It often was simply the Washington art of networking or practicing politics during an election.

Mr. Mueller’s mandate was to investigate “any links” between a Trump colleague and Russia, a broad order that had him tracking down seeming innocuous encounters.

“There are over a dozen Trump associates never accused of any wrongdoing yet mentioned more than 100 times each in often misleading and pejorative ways,” said a former Trump adviser who asked not to be named for fear of receiving a Democratic subpoena.

This person wants Republicans to ask why Mr. Mueller violated the spirit of Justice Department guidelines. The former special counsel is to appear before two panels — the House Permanent Select Committee on Intelligence and the House Judiciary Committee — in open session. He is scheduled for a total of four hours of open testimony, followed by closed-door sessions.

“Mueller violated the civil rights of President Trump and dozens of his associates,” the former adviser said. “It was a crime wave of defamation.”

J.D. Gordon, a former adviser, has said his role in a Republican convention platform amendment on Ukraine was skewed by omitting penitent facts.

The Mueller report devotes an index headline about Mr. Gordon being invited to breakfast by the Russian ambassador in Washington. A check of that small section finds that Mr. Gordon didn’t attend.

John Dowd, Mr. Trump’s former defense attorney, said the report misrepresented his discussions with counsel for former National Security Adviser Michael Flynn.

Carter Page, an Oklahoma City-based energy investor, was one of the most targeted Trump advisers. The FBI tapped his communications for 12 months in 2016 and 2017. It also assigned a spy, Mr. Halper, who engaged him in conversations at Cambridge University and at Mr. Halper’s home in Virginia.

In the end, Mr. Page faced no charges.

Former British spy Christopher Steele, in a Democratic Party-financed dossier written from June to December 2016, accused Mr. Page of a number of felonies. Clinton operatives spread the false charges among journalists, and the FBI used the dossier extensively. The entire document was sourced to Kremlin operatives whose motives to this day are not known.

In a footnote, the Mueller report refers to Foreign Intelligence Surveillance Act (FISA) warrants on Mr. Page. But the report doesn’t point out that the evidence the FBI presented to judges was largely based on the discredited dossier.

Two Republican senators in January 2018 sent a letter to the Justice Department accusing Mr. Steele of lying by telling the FBI he had not briefed reporters when in fact he had.

A House Republican investigation last year showed that the FBI never confirmed Mr. Steele’s conspiracy allegations and gave him low marks for his Russia information.

Republicans plan to ask Mr. Mueller why he devoted no report space to the dossier, given that it had a such a broad impact on the Russia investigation.

The Justice Department inspector general has been investigating how the FBI used the dossier.

As he left his post on May 31, Mr. Mueller stated emphatically that he would not speak beyond the words he put into his report.

Mr. Caputo is one of the most conspicuous Trump investigation critics. At one point, he denounced the Senate Select Committee on Intelligence with “God damn you to hell” for putting innocent Republicans through hours of questioning and legal bills.

House Democrats have floated conspiracies around Mr. Caputo, hung on the fact that he briefly lived in Russia as a post-Soviet proponent of democracy. He has never been accused of any wrongdoing.

Mr. Caputo sat down with the Mueller team early in its May 2017 creation.

“My interviewer seemed to believe there was Russian collusion,” he said. “For three hours he asked me Russian-flavored questions. … I gave them nothing.”

Mr. Caputo found his name on Page 61 in a few paragraphs about being approached by a Russian.

“Absolute bold omissions and really slutty innuendo,” he said.

He told The Times that Mr. Mueller’s staff of Democrat-aligned prosecutors omits what he considers a fact: that the Russian national who approached him in May 2016 was a longtime FBI informant. Mr. Caputo says his hired private investigator confirmed this.

“This report is garbage,” he said.