Britain and France are expected to argue that current sanctions are not working and call for the embargo to be altered at a meeting with European Union foreign ministers in Brussels.
Mr Hague's calls look set to be rejected with several EU countries - including the Netherlands, Sweden and the Czech Republic - opposed to the plans over fears the weapons could fall into the hands of Islamic extremist groups.
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Michael Fallon has received reassurances from his US counterpart that security ties with Britain will "endure" following a vote for Brexit.
On Friday, police said it appeared the girl had shot herself in the face with the weapon.
The showers will be slow-moving, heavy and thundery at times, particularly in central and eastern areas.