Sources at the contractor confirmed to the Enquirer that it will carry out the work for property owner Blackstone.
The FCA moved into its new east London offices this summer.
The 15-storey building at 25 North Colonnade will now be completely revamped by ISG to designs drawn-up by architects Gensler.
It is understood ISG fought-off competition from BAM and Sir Robert McAlpine to win the job.