The envoys held talks behind closed doors at the UN Security Council offices to discuss the resolution after the suspected use of chemical weapons near Damascus last week in an attack which left hundreds dead.
Britain wants the resolution to allow "all necessary measures to protect civilians," which could mean military action.
Russia fiercely opposes giving UN backing to what it considers western moves to oust Assad.
Russia has said the Council should wait for UN chemical weapons inspectors to finish their inquiry in Syria before deciding any action. UN leader Ban Ki-moon said today that the inspectors need four more days to finish their work.