The contract for the lodges has been tendered separately from the main construction package on the £250m development which has gone to Bowmer & Kirkland.
The Enquirer understands Kier is now on a shortlist of two for the job and is favourite to win.
The original bidders also included Ardmore, ISG and Leadbitter.
One source close to the project said: “Kier are very keen to win this.
“They have put in a very competitive bid and the word is their price is the lowest by a fair margin.”
The new site will be the fifth holiday village to be built by Center Parcs in the UK.