"Obama authorised the whole thing. The only thing you saw announced in a press release was the 21,000," a defence official familiar with the troop-approval process told the Washington Post.
The report, posted on the newspaper's website late on Monday, comes as Obama weighs a request from General Stanley McChrystal, the US and Nato commander in Afghanistan, for more combat, training and support troops, with several options including one for 40,000 extra soldiers.
But the newspaper noted that the maximum number of US service members expected in Afghanistan by year's end - 68,000 - would remain the same.
Major deployments of support troops have not been publicised by the Pentagon and the White House in the past.
When George Bush, the former US president, announced a US troop increase in Iraq, he only mentioned 20,000 combat troops and not the accompanying 8,000 support troops.
The troop increase approved by Obama brought the level of US forces deployed in the Iraq and Afghanistan theatres to a total greater than during the peak of the surge in Iraq in late 2007 and early 2008.
At the start of this month, some 65,000 US forces are currently in Afghanistan and about 124,000 in Iraq, compared to around 26,000 US troops in Afghanistan and 160,000 in Iraq at the height of the Iraq "surge", according to a troop count by the Washington Post.