Yes, it is possible to use the ArcPy library on a Linux operating system, given that either ArcGIS for Server or ArcGIS Engine are installed. This automatically installs the ArcPy site package.
However, the ArcPy library is not available as a stand-alone Python installation in Linux prior to 10.5, because ArcPy requires that the binaries are installed together with either ArcGIS Engine or ArcGIS Server. Python scripts executed in a Linux shell prior to 10.5 need to be invoked against the Python 2.7 runtime that is shipped with ArcGIS Server.
Beginning at 10.5, select tools can now be executed with a Python 3.5 runtime that is installed independently of ArcGIS Server. This allows a more native Linux experience when writing Python scripts, including the use of third-party modules and native Linux paths. For details, refer to ArcGIS Server: Python 3 runtime for ArcGIS Server on Linux.
Note: Since ArcGIS for Desktop is not supported on a Linux operating system, a script written in ArcGIS for Desktop may not work.
Warning: Do not modify or upgrade the Python version installed with Esri products.