The Obama administration is considering sending American military trainers to Syria to work with the rebel fighters in an expansion of the CIA's training program already underway in Jordan.

A U.S. official tells Fox News the prospect of sending trainers to Syria is being “kicked around” at the White House, but is part of the administration’s broader Syria policy and would not be tied to a potential military operation.

The official stressed that one of the major obstacles in launching this type of operation is that they are having serious problems vetting which are the trustworthy groups in Syria.

The proposal to use the U.S. military to train the rebels -- something the administration has resisted through more than two years of civil war -- would answer the demands of some lawmakers, including Sen. John McCain, R-Ariz., to do more to train and equip the Syrian opposition.

The CIA has been training select groups of rebels in Jordan on the use of communications equipment and some weapons provided by Gulf states. The new discussions center on whether the U.S. military should take over the mission so that hundreds or thousands can be trained, rather than just dozens.

Any new training program conducted by the U.S. military would take time to put in place and likely would not begin until after any potential military action had been taken regarding the chemical weapons attack. The Pentagon already has at least 1,000 troops in Jordan, including trainers working with Jordanian forces. The U.S. left about a dozen fighter jets and a Patriot missile battery there after a recent training exercise.

Gen. Martin Dempsey, chairman of the Joint Chiefs of Staff, has told Congress that the U.S. military would be prepared to do more training for the Syria opposition if needed.

The Associated Press contributed to this report