We are using t4 templates for managing configurations in our project. And web.config file is generated via web.tt file. After generation in csproj file I have following:
<Content Include="Web.config">
<AutoGen>True</AutoGen>
<DesignTime>True</DesignTime>
<DependentUpon>Web.tt</DependentUpon>
</Content>
Is it possible to configure t4 template somehow to generate independent web.config file, not under web.tt?

Yes you can. First define a save routine in a T4 include file.
SaveOutput.tt:
Now every time you call SaveOutput with a file name it will write the current buffer to that external file. For example.
Web.tt:
If you want to generate multiple files: