Bam is in final discussions with the University of Sheffield to deliver its new social sciences building.
Kier had been preparing the site but a university spokeswoman said the institution had been unable to agree a price with the company and therefore discussions with Bam had begun.
She said: “Following negotiations, the University of Sheffield has unfortunately been unable to agree a price with Kier for this vital new construction project.
“Kier will complete the site preparation works which are already under way and will remain a valued member of our approved supplier list for other university work.”
She added: “We have identified an alternative contractor, Bam, and are in the process of finalising an agreement with them to deliver this project.”
The new 16,500 sq m social sciences building is due to be completed in time for the 2021/22 academic year.
A spokesman for Bam confirmed the company is in discussions with the university.
Kier has been approached for comment.