House Oversight and Government Reform Committee Chairman Darrell E. Issa, California Republican, and regulatory affairs subcommittee Chairman Jim Jordan, Ohio Republican, said they have discovered that the head of the investigation is Barbara Kay Bosserman, a trial lawyer in the Justice Department who donated more than $6,000 to Mr. Obama’s 2008 and 2012 campaigns, as well as several hundred dollars to the national Democratic Party.

“The department has created a startling conflict of interest,” Mr. Issa and Mr. Jordan said in a letter sent Wednesday and reviewed by The Washington Times. “It is unbelievable that the department would choose such an individual to examine the federal government’s systematic targeting and harassment of organizations opposed to the president’s policies.”

The Internal Revenue Service’s internal auditor revealed in May that the agency had been inappropriately targeting and blocking applications for tax-exempt status from tea party and conservative-leaning groups. In the immediate aftermath, Mr. Obama promised that the FBI and the Justice Department would investigate whether the IRS broke any laws.

Eight months later, the investigation has not produced any public findings, and Mr. Issa says the FBI and Justice Department have stonewalled his efforts to find out what’s going on.

Ms. Bosserman didn’t respond to an email seeking comment.

But the Justice Department said it isn’t allowed to consider a career lawyer’s political leanings when doling out assignments and that it would violate an employee’s constitutional rights if he were penalized on the job for making legal political contributions.

“It is contrary to department policy and a prohibited personnel practice under federal law to consider the political affiliation of career employees or other non-merit factors in making personnel decisions,” said spokeswoman Dena Iverson.

According to campaign finance records, Ms. Bosserman donated $400 to the Democratic National Committee in 2004 and $250 in 2008. She gave $3,600 to Mr. Obama’s campaign in 2008, $2,000 to his campaign in 2012, and $500 to the separate Obama Victory Fund in 2012.

The tea party scandal has faded from the headlines but the fallout continues. The IRS is trying to come to terms with some of the conservative groups it delayed in approving tax-exempt status.

Meanwhile, the criminal investigation continues, according to an FBI letter sent to Mr. Issa late last month. The FBI says it’s because of that investigation that the agency will not release any of its documents to Mr. Issa.

“We would request that the committee permit the investigators to complete their investigation and consult with federal prosecutors,” Stephen D. Kelly, assistant director of the FBI’s office of congressional affairs, said in a Dec. 31 letter to Mr. Issa. “As a result, we cannot provide the documents requested … while the criminal investigation is active.”

The FBI didn’t say when it would complete the investigation.

Mr. Issa and Mr. Jordan warned Attorney General Eric H. Holder Jr. that the FBI’s refusal to cooperate could rise “to the level of criminal obstruction” of Congress’ oversight responsibilities. The Justice Department promised to at least brief congressional investigators on the status of the FBI probe, but then backed out.