The ban has caused considerable controversy, with critics alleging the US government was trying to hide the human cost of its wars in Iraq and Afghanistan.
Robert Gibbs, the White House press secretary, said the decision gave families the final say and permitted them "to make that decision and protect their privacy if that's what they wish to do".
The president was supportive of the secretary's decision, Gibbs added.
The move was welcomed by some veterans groups, who said that the deaths should not be "hidden from view".
"The sight of flag-draped coffins is, and should be, a sobering reminder to all Americans of the ultimate sacrifice our troops have made and the high price of our freedom," Paul Rieckhoff, executive director of the Iraq and Afghanistan Veterans of America, told AP on Thursday.
However Meghan Tisinger, spokeswoman for Families United for Our Troops and Their Mission, said the reported move showed "a complete disregard for the will of America's military families and the need for their privacy during this solemn moment."
There have been several legal challenges to the policy, with the Pentagon in 2005 releasing hundreds of military images of flag-draped coffins of soldiers killed in Afghanistan and Iraq following a freedom of information request and lawsuit.