The Queensferry Crossing had been expected to open at the end of 2016 but there are fears that adverse weather conditions may delay its completion.
Transport Scotland said the contractor was working to ensure the Â£1.35bn bridge was open "as soon as possible".
The Scottish Conservatives said any delay would be of "serious concern" to commuters.
In February, the national transport agency said the project was scheduled to finish "on time and under budget".
But in the wake of newspaper reports that the opening of the bridge could be delayed, they would not name a completion date.
Nicola Sturgeon is to field a question on the matter on Thursday, with Lib Dem MSP Alex Cole-Hamilton lodging a query about the opening date at first minister's questions.
A Transport Scotland spokesman said: "As previously stated the contractor, Forth Crossing Bridge Constructors, is making every effort to open the bridge as soon as possible, despite the challenge that the weather presents."
Alex Johnstone, transport spokesman for the Scottish Conservatives said the government "needs to be open" so commuters know when the bridge will open.
"Ministers weren't slow to celebrate the bridge coming in under budget, and have said for some time that it is on schedule," he said.
"The public will expect them to be equally open when things aren't going to plan."

After the best part of a century of service, the traditional steel "lattice" pylon has been updated.
The "T-pylon" is shorter, standing at about 120ft (36m); the old steel giants are typically 165ft (50m).
The National Grid says it will respond to the need to harvest energy from an increasing number of lower-carbon energy sources.
The new tower can be shorter yet still capable of operating at 400,000 volts because of the way the cables are held in place.
Instead of being attached to three arms, a diamond arrangement is used to carry the cables off in one arm in a much smaller space.
Each arm has to carry 60 tonnes. With only eight main structural components plus bolts, it is easier to erect and install - taking a day rather than a week.
According to the National Grid, new pylons are needed to respond to the move away from coal and towards other forms of generation such as wind, solar and nuclear.
These new low-carbon energy sources come from different geographical locations to the "traditional ring" of coal-fired generation in the centre of England.
This test line won't be connected to the rest of the grid, it will be used to train staff and contractors.
In particular, it will be used for people to practise "stringing" the conductors (wires) on to the pylons, as a very different technique is needed.
The T-Pylon design was adopted after an international competition held by the National Grid in 2011, won by Danish company Bystrup.
The claim is that these pylons will be less obtrusive not just because of their shorter stature, but because the design allows them to "follow the contours of the land".
According to project manager Alan Large, maintenance will be easier because operators will not have to climb up the tower, they will work from elevated platforms positioned alongside it. Their smooth, impenetrable surface will also make them more difficult to vandalise.
It won't replace the 88,000 lattice pylons that currently bestride the UK countryside. It will principally be used in the construction of new power lines in England and Wales.
The National Grid is applying to use them to connect the new Hinkley Point C nuclear power station to the UK's electricity transmission network.
The start of the UK's electricity transmission network began with the building of the first pylon in 1928 near Edinburgh.
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To some they are a blot on the landscape; to others they have a kind of skeletal beauty.
But for a small number of pylon enthusiasts, these giant structures hold a lifelong fascination.
Read more: Pylon passion

16 September 2014 Last updated at 07:25 BST
Leaders from both sides have been touring the country and encouraging people to vote.
Scotland's first minister Alex Salmond, who is leading the campaign for the "Yes" vote, tells Newsround what he thinks being Scottish means.
The Scottish government has been urged to confirm when the new Forth bridge will open.
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The first new design of an electricity pylon in almost 90 years has been erected at a site in Nottinghamshire.
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There are only a few days left until the people of Scotland decide whether or not to become an independent country.