The District’s attorney general moved Thursday to settle a dispute with federal prosecutors over documents they want to review as a part of their long-running investigation into Mayor Vincent C. Gray.

In a letter to U.S. Attorney Ronald C. Machen Jr., Attorney General Irvin B. Nathan proposed having a federal judge review the materials, which pertain to the District’s controversial 2011 settlement of a $15 million claim by D.C. Chartered Health Plan, then the city’s largest health-care contractor.

The firm is owned by Jeffrey E. Thompson, who is alleged to have funded a secret $655,000 effort to elect Gray (D) in 2010. Thompson has not been charged, but several people familiar with the probe say he is the unnamed executive described in court documents as funding the “shadow campaign.”

Thompson had unsuccessfully raised his claim with officials during the administration of Gray’s predecessor, Adrian M. Fenty (D). Eight months after Gray took office, his administration settled the matter for $7.5 million. Prosecutors appear to be interested in whether there is a connection between the deal with Thompson, which city officials have defended as well justified, and his support for Gray’s campaign.

Nathan has declined to share the documents sought by investigators, which he says in the letter are protected by “the very heart” of the attorney-client privilege. Machen, in a letter sent to Nathan on Wednesday, asked the city to waive that privilege.

Neither party would share the letters exchanged, but Nathan’s office issued a lengthy news release describing their contents Thursday afternoon, a day after The Washington Post first reported the dispute.

Under Nathan’s proposal, the documents in question would be given to Richard W. Roberts, chief judge of the U.S. District Court in Washington. Roberts, who oversees the grand jury involved in the probe, would review the documents and determine whether the attorney-client privilege applied.

If Roberts were to rule the materials are privileged, they would be returned to the city. Nathan pledged to make any non-privileged documents public “so that everyone can see, as I have repeatedly advised, that there is nothing remotely incriminating in them.”

Any agreement between the parties would be subject to Roberts’s approval, according to a court official.

Machen’s office said it is reviewing Nathan’s letter and urged District leaders to reconsider withholding documents.

“Given the strong public interest in uncovering illegality among government officials, the citizens of the District would be best served by a timely and unfettered investigation rather than protracted litigation,” the office said in a statement.

The city’s statement noted that Nathan’s office has provided nearly 20,000 documents and numerous interviews pertaining to the campaign investigations. It also said that Nathan had previously offered to submit to a “no holds barred” interview with investigators and to release documents if Machen could offer evidence of illegal acts connected to the settlement.

“If there were any criminality afoot in the settlement decision, one would have to conclude that a half dozen sworn officers of the court did not see it or willfully participated in it,” Nathan wrote in the letter. “I am unable to come to that conclusion, and I do not think your office would either.”

The materials Machen is seeking, Nathan said, consist of internal correspondence between city attorneys and officials in the Department of Health Care Finance, including discussions of “litigation and negotiating tactics.”