I can find a lot of obsolete threads but a lot have been changed the last months.
Is there currently any possibility to run .NET Core and/or ASP.NET Core on Raspberr
You can get a (seemingly) stable build of .Net Core on Raspberry Pi 2 or 3 by following Peter Marcu's instructions here: https://github.com/dotnet/core/blob/master/samples/RaspberryPiInstructions.md
You don't need to use "corerun", you can use "dotnet". You build on your dev machine and copy to your Pi.
Update: I realise now the original question specifically mentioned Rasbian (Jessie). The method at that link requires Ubuntu 14.04 or 16.04. But if that isn't a problem I think you should be good to go.