This is expected "over the coming months" once driver training has been completed.
The introduction of the Japanese-built Hitachi trains has been delayed for months because drivers said they could not see signals clearly through the curved windscreens.
A new flat windscreen has been fitted to one of the trains, which ScotRail said had been tested successfully this week.
The first of the class 385 trains should have started carrying passengers on ScotRail's flagship route last September.
Scotland on Sunday, The Scotsman's sister paper, revealed the windscreen problem in February.
Perry Ramsey, operations director of the ScotRail Alliance with track body Network Rail, said today: "We have trialled the modified driver's windscreen on the new class 385 trains this week.
"The feedback from all parties including [train drivers' union] Aslef was that the flat windscreen manufactured by the existing supplier would be fit for purpose.
"The position of the windscreen proving has been deemed successful using this flat windscreen and the issue of multiple images has been resolved.
"As a consequence, ScotRail intends to start the process of driver training and introducing the class 385 into passenger service over the coming months."
ScotRail has been forced to lease a stand-in fleet of ten trains because of the delay, which are expected to start carrying passengers from July.
This is because some of its other trains have been transferred to other operators after their leases expired.
A total of 70 class 385s have been ordered, which will also run on lines across the Central Belt, including to Dunblane and Alloa once electrification of those lines has been completed.
Aslef Scottish organiser Kevin Lindsay said: “I’m happy a practical solution has been found to fix the sighting issues on these trains.
"We will work with ScotRail to get these trains introduced as quickly as possible.
"However, questions must be asked as to why [ScotRail operator] Abellio and [Scottish Government agency] Transport Scotland allowed Hitachi to manufacture a train with such a serious issue such as a driver being unable to see the signals clearly.
"Any additional cost associated with this refit should not be met by the Scottish taxpayers.
"It’s no wonder the Scottish people back a nationalised rail industry when the private sector can’t even get a windscreen right.”
A ScotRail Alliance spokesman said: "Good progress has been made to resolve the problem with the windscreen.
"We will continue to support Hitachi, and work with our trade union colleagues, to get these brand new electric trains into service as soon as possible."