Unable to install MySql.Data in Visual Studio using nuget
With a design by NuGet v3. If one package source is invalid or is failing then it is not possible to install or update a NuGet package when all package sources are being used. So you need to check the package source from Tools -> Options -> NuGet Package Manager -> Package Sources first. Make sure the package source is correct. (you can unable other package sources except nuget.org). The source of nuget.org is "https://api.nuget.org/v3/index.json"
Or you can try to add another one with the URL on API v2 of the NuGet.org:
The source of nuget api v2 is "https://www.nuget.org/api/v2/".

Besides, if above not help you. Please check you NuGet Packager Manager version: Go to VS menu Tools > Extensions and Updates > Updates and check if there is no updates for NuGet Package Manager. Installing the latest version.
Below method may be optional:
In Explorer open folder where your project resides. Open packages.config using Notepad. Find and remove the line that mentions corrupted package name. Then Open folder where your solution resides. Open subfolder "packages". Find folder with corrupted package and remove it. Try to install that package again.
Hope this help.