It is understood they would involve major displays featuring both science and European art and design.
Bosses see the extension to the Chambers Street museum as a key part of its masterplan.
It is hoped the new galleries will be in place by 2016, with a further three to be added by 2020, taking the total spend on the project to £70 million.
Nearly 1.5 million visitors have come through the doors since its overhaul in July.
Dr Gordon Rintoul, the museum’s director, said: “The next phase of the masterplan will sustain our success and ensure that the museum continues to develop as a ‘must-see’ destination for Scottish and international visitors, and a source of inspiration and wonder for generations to come.”