whether .Net Core(CoreCLR) supports C++\CLI (clr\oldsyntax) runtime environment ?
As far as I know there is no plan to support C++/CLI with .NET Core.
If no, what can be the possible solutions to this application work ?
You can (should) provide a C API. Mono e. g. supports P/Invoke and .NET Core also supports P/Invoke (see also this Stack overflow question and this DllMap related ticket).
Update (2022-09-02): This answer is from 2016. See the other answers (e.g., this) for what is possible with recent .Net Core versions.