The upgrade is also thought likely to improve reliability and could enable more frequent trains on the Edinburgh-Tweedbank line.
The latest ScotRail electric trains could run on the route to replace the current carriages which are among the operator’s oldest and least reliable.
These have the potential for reducing the hour-long journey by being able to accelerate faster from stations and up to the 880ft Falahill summit, south of Gorebridge, which is the tenth highest in Britain.
Adding overhead electric wires to the 35-mile line would also be much easier than on many routes because it was “future-proofed” for such a development.
Space was left under bridges and in tunnels for the wires to be added when the railway was built four years ago.
The Scottish Government’s Transport Scotland agency confirmed the scheme was being examined.
A spokesperson said: “We remain committed to a rolling programme of electrification, and are working with Network Rail to identify the next Scottish electrification scheme.
“Several routes are under consideration.
“Electrification of Scotland’s railway network is a key component of the Scottish Government’s commitment to improving rail services and tackling climate change.”
A Network Rail spokesperson said: “We are working closely with the Scottish Government to identify opportunities to extend electrification and the faster, more reliable and greener journeys it brings.”
The Campaign for Borders Rail welcomed the news but said other improvements were also needed. Chairman Simon Walton said: “Positive news of electrification, no matter how early in the process, is very encouraging for the railway, and for our ambition to see the line extended through Hawick to Carlisle.
“It’s better for the railway, particularly given the challenging topography. Electrification would offer better reliability and capacity, with newer and faster trains.
“However, we must continue to campaign vigorously for further infrastructure enhancements – such as more double track sections and a better junction with the east coast main line at Portobello in the east of Edinburgh.”
David Spaven, author of Waverley Route, said: “Electrification – with faster acceleration of trains – would help to overcome the significant constraints imposed when the Scottish Government decided to build the line with much less double track than previously planned.”