Wi-fi link and easier e-mail
Ministers are expected to ask First ScotRail for a detailed study into the move, to attract more business people to travel by train.
National Express East Coast already offers free wireless internet connections on its Scotland-London trains on the east coast main line.
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A trial is also being launched on the Glasgow Subway, starting with Buchanan Street station in the city centre.
It is thought that the Edinburgh- Glasgow scheme could improve reception for passengers making phone calls as well as aiding those wanting to send e-mails and use the internet from computers.
Scottish Citylink introduced wi-fi connections for its coach passengers travelling between the two cities last year. First ScotRail has previously said it was aware of increasing interest in wi-fi among business people and has been in talks with the Scottish Government's Transport Scotland agency.
Other improvements being considered as part of First ScotRail's three-year franchise extension to 2014 include upgrading the London-Scotland sleeper services, and more trains during the Commonwealth Games in Glasgow in 2014.