North Yorkshire County Council has awarded preferred bidder status to Sisk pending contract award.
This is dependent on funding from the Department for Transport which is currently reviewing the Full Business Case with a decision expected in October.
Following tender returns, the estimated cost of the project has increased by £7.2m to £68.8m in the face of cost inflation headwinds.
Sisk will move on site this December to clear the moorland route by late February 2023 to avoid the bird nesting season.
An estimated completion date is May 2025.
Kex Gill, between Skipton and Blubberhouses, has a history of landslips and over the years there have been many unplanned and costly closures.
They cause an inconvenience for users of the A59, the surrounding towns and villages, and are damaging for businesses and local farmers.
Sisk now has a strong local presence in Yorkshire having recently completed the Leeds Public Transport Investment Programme City Centre Gateways and is delivering major infrastructure work at York Central.
Dominic Hodges, managing director UK Civils at Sisk, said: “The scheme is technically challenging with complex engineering solutions required within an environmental sensitive area.
“We are working collaboratively with all stakeholders to develop an innovative scheme solution and look forward to working with the local community to leave a lasting legacy.”