The development, for Housing 21, will provide 80 apartments and 24 stand-alone bungalows for older people to live independently but have access to on-site care and support when needed.
Work is expected to complete in the summer of 2021.
The Oxenholme project is the second extra care scheme that Engie is building for Housing 21, with work on a similar £10m scheme in Bentham (on the Yorks/Lancs border) reaching the halfway mark.
In addition to this, Engie has another three extra care building projects in Cumbria set to start in the coming months, with a value of approximately £60m.
Engie area director Lee Francis said: “Engie has a long track-record of creating high-quality extra care developments and we are pleased to be continuing our partnership with Housing 21 and making this scheme a reality.”