In the memo, Mr. Barr established a series of requirements governing whether investigators could open preliminary or full “politically sensitive” criminal and counterintelligence investigations into candidates or their donors.

No investigation into a presidential or vice-presidential candidate — or their senior campaign staff or advisers — can begin without written notification to the Justice Department and the written approval of Mr. Barr.

The F.B.I. must also notify and consult with the relevant leaders at the department — like the heads of the criminal division, the national security division or a United States attorney’s office — before investigating Senate or House candidates or their campaigns, or opening an inquiry related to “illegal contributions, donations or expenditures by foreign nationals to a presidential or congressional campaign.”

Past attorneys general have said that the department must take extra care with politically sensitive campaign-related investigations in an election year. But Mr. Barr is the first to require that the F.B.I. consult with the Justice Department before opening politically charged investigations.

Since he took office, he has handled a seemingly endless series of politically charged investigations and their aftermaths, and he has drawn rebukes for how he has done so.

Mr. Barr faced withering criticism over his delivery of the Mueller report to Congress. While he allowed the report to be made available to the public with only light redactions, Democrats and some centrist Republicans accused him of playing down the severity of the findings.

The attorney general is also overseeing a criminal inquiry into the decision to open the investigation into Mr. Trump’s campaign.