The official, whose country is part of the U.S.-led NATO alliance in Afghanistan, requested anonymity as he was not authorised to speak with journalists.
An integral part of Resolute Support has been to train and equip Afghan security forces fighting the Islamist Taliban, which was ousted from power in 2001 and has since waged an insurgency.
"We have made an offer but it is for authorities in Qatar to decide if they are comfortable with NATO using their territory as a training ground," said a second security source based in Washington DC.
A third source, a diplomat based in Kabul, said bringing "Afghan special force members to Qatar for about four to six weeks of rigorous training" was under discussion.
Qatar's government and NATO's communications office did not respond to questions about the proposal to use the Gulf state as a base for training Afghan forces. The Afghan government also did not repond to a request for comment.
UPSURGE IN FIGHTING
Around 7,000 non-U.S. forces, mainly from NATO countries but also from Australia, New Zealand and Georgia, outnumber the 2,500 U.S. troops left in Afghanistan.
The final exit of foreign forces comes amid a surge in fighting between Taliban fighters and Afghan forces in several provinces.
Fears the Taliban could over-run Afghanistan's battered security forces, which have relied heavily on NATO support, intelligence, and logistics - particularly U.S. air support - have been heightened in recent weeks as the militants have launched major offensives, seizing districts and overrunning military bases.
Earlier this month, NATO Secretary-General Jens Stoltenberg said the alliance was "looking into how we can provide out-of-country training for the Afghan Security Forces, especially the Special Operations Forces".
Qatar, an energy-rich Gulf state has been home to the Taliban's political office since 2013. In recent years, this has been the only known venue where authorised representatives of the hardline insurgent group have held talks with U.S. officials, representatives of NATO, international rights groups and Afghan government officials.
Two sources said the United States, Britain and Turkey were among the NATO countries ready to send a force to train Afghans in Qatar.
A Taliban spokesperson said the group was not aware about NATO's plan to train Afghan forces in Qatar.
"In the case of Afghan soldiers who receive military training abroad... If peace is established then maybe the well-trained should be hired to serve Afghanistan but if they come and fight against us and their nation, then of course they will not be trusted by us," said Zabihullah Mujahid, a Taliban spokesperson.
(Additional reporting by Orooj Hakimi in Kabul, Charlotte Greenfield in Islamabad, Robin Emmott in Brussels; Editing by Alex Richardson) ((Rupam.Jain@thomsonreuters.com; +91 7042133028;))