Managing director Alex Hynes said that was the earliest that enough trains would be available, but their introduction then could not be guaranteed because of the complexity of the testing process.
The Japanese-designed Hitachi Class 385 trains were due to have been introduced on the route this autumn.
They have been delayed by problems with electrifying the line and building the trains at a Hitachi factory in County Durham.
The Conservatives said the delay was "hugely disappointing" to commuters.
Mr Hynes told Holyrood's rural affairs and connectivity committee the other electric trains would start running on the line from next month as planned - although it is a year later than scheduled.
Spare Class 380 trains - which operate on lines such as Ayr and North Berwick - will take over some peak-hour services from early December.
The ScotRail chief said he expected to have 21 of the new Class 385 fleet trains available in February.
However, he added: "I'm not yet prepared to make a commitment [over introduction] because of the complexity of what we are trying to achieve.
"I can't give you a cast-iron guarantee because it's not a process I am wholly responsible for" - since Hitachi and regulator the Office of Rail and Road are also involved.
He also repeated his view that "rushing them into traffic" before being fully tested could hit the route's performance, which was "the last thing customers want".
They will run with up to eight carriages compared to the current maximum six and reduce journeys by ten minutes to 42 minutes from December next year.
However, Mr Hynes said there were no plans to run a non-stop 30-minute service between the two cities.
He said: "The correct balance between capacity and journey time is probably right, but we will keep it under review."
Scottish Conservatives transport spokesman Jamie Greene, who sits on the committee, said: "Whilst we accept some of the testing functions are outside of the control of ScotRail, and testing is an essential part of implementation, there still seems to be no guarantee as to when the Class 385s will be fully running on this busy and important line.
"This is hugely disappointing news for Central Belt rail commuters.
“There seems to be a constant blame game between Hitachi, ScotRail, and now these “external factors” which will affect testing and implementation.
"Humza Yousaf is the transport minister and he really needs to get a grip over delivery of these long overdue trains."
Scottish Labour transport spokesman Neil Bibby said: “The SNP Government and ScotRail gave a clear promise to passengers that these new trains would be operational months ago.
"Passengers travelling between Glasgow and Edinburgh have had to put up with delay after delay and it is unacceptable.
"Scotrail have told Parliament the new trains may be fully operational by February but have failed to give a clear guarantee that they will be in service by then.
"Passengers will again be concerned - not giving a guarantee is very telling and simply not good enough."