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."
The Scottish government has been urged to confirm when the new Forth bridge will open.