The report said Allen's return to Afghanistan to resume his responsibility as the top commander of coalition forces was confirmed Wednesday by a Pentagon spokesman.
Allen had been away from Afghanistan following a Pentagon announcement it was investigating the "potentially inappropriate" communications between him and a woman in Tampa, Fla., the report said.
The probe also relates to David Petraeus, Allen's predecessor in Afghanistan who went on to become director the U.S. Central Intelligence Agency. Petraeus resigned his position earlier this month because of an extramarital affair.
The Hill reported Allen has denied any wrongdoing and earned the support of U.S. Defense Secretary Leon Panetta, who wants him to keep serving in Afghanistan while the Pentagon probe continued.
Earlier, the report said Allen had been nominated to be NATO's Supreme Allied Commander-Europe and to head the U.S. European Command, but that must wait while the investigation is progressing.
Khaama Press quoted a statement from Allen's Kabul office that he was happy to be back in time to celebrate Thanksgiving with his troops.
"The Defense Department inspector general's investigation into certain communications by Gen. Allen continues," the report quoted the statement as saying. "Out of respect for that process, Gen. Allen will continue to refrain from commenting on those matters that may fall within the scope of the investigation."