The senior US defense official, who spoke on condition of anonymity, said the decision would also allow greater use of US air power, particularly close air support.
However, the official cautioned, "This is not a blanket order to target the Taliban."
Obama's decision comes more than a year after international forces wrapped up their combat mission in Afghanistan and shifted the burden to Afghan troops.
It also comes ahead of Obama's eagerly anticipated decision on whether to forge ahead with a scheduled reduction in the numbers of US troops from about 9,800 currently to 5,500 by the start of 2017.
A group of retired generals and senior diplomats urged Obama last week to forgo those plans, warning they could undermine the fight against the Afghan Taliban, whose leader was killed in a US drone strike in Pakistan last month.
Under the new policy, the US commander in Afghanistan, General John Nicholson, will be able to decide when it is appropriate for American troops to accompany conventional Afghan forces into the field -something they have so far only been doing with Afghan special forces, the official said.