Investigators last month indicated that the flight was diverted deliberately and its communication systems manually switched off as it was leaving Malaysian airspace, triggering a criminal investigation by police that has revealed little so far.
The fate of flight MH370 has been shrouded in mystery, with a number of theories put forward including a hijacking or terrorist attack and a pilot gone rogue.
There have been unconfirmed reports in the Malaysian media of calls by the captain before or during the flight but no details have been released.
The NST report said that Flight 370 flew low enough near Penang island on Malaysia's west coast — after turning off-course — for a telecommunications tower to pick up the co-pilot's phone signal.
The phone line was "re-attached" between the time the plane veered off-course and blipped off the radar, the Government-controlled paper quoted the second source as saying.
"A 're-attachment' does not necessarily mean that a call was made. It can also be the result of the phone being switched on again."
Malaysia's Transport Ministry told AFP that it was examining the NST report and would issue a response.
The Malaysian Government and media have repeatedly contradicted each other and themselves over details of the search and criminal investigation.
- AAP