The 50-year-old man was taken to hospital where he is being treated for burns to his face.
It is thought he was helping to erect a lamppost at the site when it hit an overhead cable.
BAM is main contractor on the £30m job to build a new signalling control centre for Network Rail.
A Network Rail spokesman told the Basingstoke Gazette: “A contractor working on Network Rail’s new control centre in Basingstoke was injured yesterday afternoon while working on the site, managed by BAM Construction.
“An investigation will be carried out and the Health and Safety Executive has been informed. Our thoughts are with him and his family.”
BAM Construction said: “We confirm that there was an accident on our Basingstoke site late on Monday afternoon, where we are working for Network Rail.
“We deeply regret this has happened, and our thoughts are with those affected.
“We are carrying out an urgent investigation into this incident, together with our client.”