The Government-owned company said it will proceed with handing out construction contracts for the road upgrade.
Three bids have been submitted, and Highways England is expected to announce its choice early next year.
A separate £60 million contract is due to be awarded late this year for supporting the management of the construction work.
We still believe our project is the best solution to the ongoing issues along the A303 past Stonehenge
David Bullock, Highways England
Archaeological fieldwork and preliminary works scheduled to start later this summer have been postponed.
Transport Secretary Grant Shapps gave the go-ahead to the £1.7 billion scheme to overhaul eight miles of the A303, including the two-mile tunnel, in November last year.
This was despite advice from Planning Inspectorate officials that it would cause “permanent, irreversible harm” to the Unesco World Heritage Site in Wiltshire.
Save Stonehenge World Heritage Site (SSWHS) challenged his decision in the High Court, and on Friday Mr Justice Holgate ruled it was “unlawful”.
There was a “material error of law” in the decision-making process because there was no evidence of the impact on each individual asset at the historic site, Mr Justice Holgate found.
He added that Mr Shapps failed to consider alternative schemes, in accordance with the World Heritage Convention and common law.
David Bullock, Highways England’s project manager for the scheme, said: “We have to wait while the Department for Transport considers its options, and in the meantime we are continuing with the process to appoint a contractor for the main works phase of the scheme.
“We have now paused our plans to carry out early, preparatory work, but the procurement process is very much live to ensure we maintain programme timescales as best as possible.
“We still believe our project is the best solution to the ongoing issues along the A303 past Stonehenge.
“It has been developed after a long and extensive collaboration with our key stakeholders, and we are still very much motivated to leave a legacy beyond the road – for Stonehenge, the World Heritage Site, our local communities and future generations.”