The crew of the foreign registered Airbus A310 were maintaining the aircraft at FL370 on air route A585 and were estimating the BIDAP position at 0234 WST. BIDAP was the boundary between the Perth and Melbourne Air Traffic Control areas of responsibilities and Perth Sector Control had instructed the crew to contact Melbourne Sector Control at BIDAP. As their aircraft approached BIDAP, the crew observed a weather return on radar along their proposed track just passed the reporting point. The crew decided to make a diversion to the north, left of track, to avoid the storm. As they were approaching BIDAP, the pilot-in-command commenced a 15 degree track change and instructed the co-pilot to advise Melbourne Sector that a diversion to the left was required. The co-pilot made a radio transmission to Melbourne Control indicating their position as abeam BIDAP and "up to 15NM left of track due weather". The aircraft was actually only 2 - 3NM off track at that time. Melbourne Control replied with an instruction for the crew to limit their diversion to 15NM due to conflicting traffic. A short time later, Melbourne Control informed Perth Control that the crew had reported 15 NM left of track due weather and established that this diversion had not been requested by the crew from either Melbourne or Perth Control. Melbourne Control were able to ensure separation from an Australian registered Boeing 747 that was on a conflicting air route. The tracks were laterally separated at the point at which the A310 crew reported their position, and the direction of travel ensured that the aircraft were diverging. The co-pilot believed he was reporting the position at BIDAP with a request for a diversion left of track and believed that the response from Melbourne Control was a clearance to conduct the diversion. Melbourne Control believed they were limiting the possibility of conflict for a diversion that had already taken place. The co-pilot's first language was not English and this may have contributed to a misunderstanding between the controller and the crew It was not established at which point the crew of the A310 commenced their diversion but, as the crew reported that it was only "a few miles prior to BIDAP", it was probably outside the lateral separation points for the two routes and, therefore, no infringement of the separation standards was likely to have occurred.