The Pentagon strongly denied the claims in the New Yorker, which cited unnamed current and former intelligence officials.
Pentagon spokesman Lawrence Di Rita issued a statement calling the claims "outlandish, conspiratorial, and filled with error and anonymous conjecture".
The story, by veteran journalist Seymour Hersh, said Mr Rumsfeld decided to expand the programme last year, broadening a Pentagon operation from the hunt for al-Qaida in Afghanistan to interrogation of prisoners at Baghdad's Abu Ghraib prison.
Seven soldiers are facing charges related to the abuse and humiliation of prisoners.
Some of the soldiers and their lawyers have said military intelligence officials told the guards to abuse the prisoners to make interrogations easier.
According to the story, the initial operation Mr Rumsfeld authorised gave approval to kill or capture and interrogate "high-value" terrorist targets.
US President George W Bush was informed of the program's existence, the officials told Mr Hersh.
"No responsible official of the Department of Defence approved any programme that could conceivably have been intended to result in such abuses as witnessed in the recent photos and videos," Mr Di Rita said.
Meanwhile, doubts over Mr Bush's Iraq policy sent his job approval rating to a record low. A Newsweek magazine poll showed it sinking to 42%, with 57% of Americans disapproving of his handling of Iraq.