The answer is yes, DXA does have plans to move to .NET Core, in line with the plans for SDL Web to move its APIs to .NET Core. These plans actually date back to February 2016 back when .NET Core 1.0 wasn't even released yet.
It seemed at that time we needed to wait for the CIL to be changed to be .NET Core compatible first, and then following that have DD4T add .NET Core support before we could change DXA to be .NET Core compatible, which is why I havn;t mentioned anything of these plans before.
Another change which looked to be flowing from the move to .NET Core seemed to be the changes in MVC 6, which would not allow our current structure of Page Controller and Region and Entity sub-controllers anymore, making the move to .NET Core a high impact breaking change.
It seems there has lately been some more clarity on ASP.NET Core and .NET Framework 4.6 (or .NET Core), which means that we could move DXA to .NET Core without needing to wait for CIL or make the high impact changes.
Currently the DXA team is busy with the DD4T & DXA merge, something which will introduce quite a few changes, that I still need to communicate to a broader audience (you will hear of it soon). I'll make sure the discussion around .NET Core support will be taken into account in this, and if it doesn't have too much impact, that it will be done as soon as possible.
Here are the first two updates on the DD4T & DXA merge I promised: