Attorneys from Thornton Law and two other firms have agreed to pay $2 million by next week to cover the cost of an investigation into whether they dramatically overcharged for their services in a class-action suit against State Street Bank and Trust that netted the lawyers more than $75 million.

At a hearing Tuesday on the planned investigation, US District Judge Mark L. Wolf said it was the first time in more than 30 years as a federal judge that he had appointed a special master. He named a former federal judge, Gerald Rosen, to look at whether Thornton, Labaton Sucharow, and Lieff Cabraser intentionally inflated their bills by double-counting lawyers and claiming exorbitant costs for dozens of low-paid lawyers, including the brother of Thornton managing partner Garrett Bradley.

Lawyers from Thornton and New York-based Labaton Sucharow admitted they overstated their legal bills in documents filed under penalty of perjury. Wolf relied on the documents, in part, to award the lawyers $75 million in legal fees and another $1.25 million for expenses.

Lawrence Sucharow, a partner at Labaton, acknowledged his firm double-counted 9,000 hours of legal work worth $4 million by mistake, calling it “very embarrassing.’’ It was “a result of miscommunication in the accounting department,’’ said the firm’s lawyer, Joan Lukey.

The mistake was caught in November only after the Globe asked about discrepancies in the bills.

Wolf grilled the lawyers about the hourly rates the three firms assigned to low-paid lawyers, called “staff attorneys,’’ who review documents. They generally earn $25 to $40 an hour, but, in court documents, the three firms listed their hourly rates at $335 to $500 an hour — often 10 times more than what they earned.

None of the organizations that sued State Street over allegations the bank had overcharged customers in international transactions have objected to their lawyers’ bills. But Wolf said that may be because the plaintiffs don’t know what happened.

“There might have been objections if class members were aware that [the fees] included double-counting and if some of the [staff] attorneys I was told had regular rates of $450 an hour were paid a tenth of that,’’ Wolf said.

Sucharow said inflated hourly rates for staff attorneys were common for law firms that represent plaintiffs and collect money only if they obtain a settlement. These firms don’t charge clients hourly rates at all, Sucharow said, but take a share of any settlement at the end.

“In essence, if the rule were you could only charge what you charge a billable client, we’d be all out of business,’’ he said.

By contrast, Wolf asked the defense lawyers who represented State Street whether they used staff attorneys and how much they pay them. William Paine, of WilmerHale, said their clients pay staff attorneys their actual hourly rate, about $35 an hour.

Wolf also asked Michael Bradley, brother of Thornton Law’s managing partner, whether he actually worked the 406 hours that were reported to the court. Garrett Bradley filed documents claiming that his brother, a criminal defense attorney not affiliated with Thornton, billed at a rate of $500 an hour on the State Street case.

“Is that correct?’’ asked Wolf.

“Yes, judge,’’ he responded, but admitted that he has never billed another Thornton client for $500 an hour. State payroll records show that Michael Bradley is often paid $53 an hour as a public defender.

Thornton Law Firm and its attorneys are already under federal criminal investigation in connection with allegations the firm reimbursed its partners and their spouses for millions of dollars in campaign contributions. On Tuesday, several federal prosecutors, including Fred Wyshak, head of the public corruption unit of the US attorney’s office, were present in the courtroom and taking notes.

But when Judge Wolf asked the lawyers, including those from Thornton, whether they would have any difficulty responding to questions from the new special master, Rosen, they said no. Evidence gathered in a civil investigation could be used in criminal proceedings, as well.

“I don’t anticipate any problems,’’ said Brian Kelly, who represents Thornton Law.

Wolf said the special master may ultimately find the fee award reasonable, or he may want to reduce it. He could also recommend sanctions if he finds misconduct, Wolf said, citing state and federal rules barring lawyers from making false statements.

Wolf delayed deciding whether Ted Frank, a lawyer with the Competitive Enterprise Institute think tank in Washington and a leading critic of excessive fees in class-action cases, can intervene in the case, as he has requested.

The plaintiffs’ lawyers opposed his involvement.

“It’s ironic, because any corporation caught telling investors something this misleading would surely face litigation from Labaton and Thornton,’’ Frank said after the hearing.

“One hopes affected investors step forward and do something about it.’’

Andrea Estes can be reached at andrea.estes@globe.com.