Another Iraqi cited in the article, Bayan Jabr, a former transportation minister, criticized The Times report for not fully quoting his response to the description in the cables of his relationship with the Iranians.

The cables said Mr. Jabr had agreed to a request from Gen. Qassim Suleimani , the commander of Iran’s Quds Force , to fly planeloads of weapons and others supplies through Iraqi airspace to Syria to help the regime of President Bashar al-Assad against American-backed rebels.

According to the cables, Mr. Jabr granted the request.

Mr. Jabr was quoted in the Times article as saying that the flights were carrying humanitarian supplies and religious pilgrims. His office restated that on Monday, and said the article should have noted that at the time, Mr. Jabr had not yet officially taken up the position of minister, and that the planes were “subject to periodic searches by the Americans.”

Another figure mentioned in the cables, and in the Times report, Salim al-Jabouri , the speaker of Iraq’s Parliament in 2014 and 2015, questioned the timing of the article and said it might have been published “to incite the street” against the government. Popular opinion in Iraq is already running heavily against the government, with protests underway throughout the country.

General al-Maksusi played down the significance of the Times report. He was depicted in the cables as putting all of Iraq’s military intelligence at the service of Iran. In comments published in the Times article, he denied working for Iran but praised the country for helping Iraq fight the Islamic State. He said he had also worked closely with the United States.

On Monday, a statement put out by General al-Maksusi’s office said that “military intelligence worked closely” with a committee made up of representatives of four countries after 2014 — Russia, Iran, Syria and Iraq — “in the fight against ISIS.”

“The details mentioned in the report did not come up with something new,” the statement said. “It’s natural that intelligence information would be exchanged when there is an agreement on intelligence sharing.”

The leaked reports were sent anonymously to The Intercept, which translated them from the Persian and shared them with The Times. The Intercept and The Times verified the authenticity of the documents but do not know who leaked them. The Intercept communicated over encrypted channels with the source, who declined to meet with a reporter. In these anonymous messages, the source said that they wanted to “let the world know what Iran is doing in my country Iraq.”

Falih Hassan contributed reporting.