Marine Corps Gen. James Conway said at a Washington news conference a ceremony marking the transfer would occur in the next few days, the New York Times reported.
Conway said Anbar no longer requires 25,000 Marines, many of whom he said would be better employed fighting the "growing bolder" Taliban insurgents in Afghanistan, the Washington Post said.
The U.S. military had planned to hand over security of the primarily Sunni Muslim province that borders Baghdad to Iraq in June, but that was postponed.
Marine spokesman Lt. Col. Chris Hughes told reporters in Anbar the number of U.S. troops in the province had dropped to 25,000 from 37,000 in February, and Iraqi police force numbers had climbed to 28,000 from 5,000 three years ago, the Times said.
Officials said insurgent attacks in Anbar are at the lowest level -- two or three a day -- since the war began in March 2003, when there would be more than 20 per day, the Post said.