Tested by applying this diff:

--- a/src/target.cpp
+++ b/src/target.cpp
@@ -1158,7 +1158,7 @@ static const AvailableLibC libcs_available[] = {
     {ZigLLVM_sparc, OsLinux, ZigLLVM_GNU},
     {ZigLLVM_sparcv9, OsLinux, ZigLLVM_GNU},
     {ZigLLVM_wasm32, OsFreestanding, ZigLLVM_Musl},
-    {ZigLLVM_x86_64, OsLinux, ZigLLVM_GNU},
     {ZigLLVM_x86_64, OsLinux, ZigLLVM_GNUX32},
     {ZigLLVM_x86_64, OsLinux, ZigLLVM_Musl},
     {ZigLLVM_x86_64, OsWindows, ZigLLVM_GNU},

And then doing the following:

andy@ark:~/tmp$ cat zigcc
#!/bin/sh
/home/andy/dev/zig/build/zig cc $@
andy@ark:~/tmp$ export CC="$HOME/tmp/zigcc"
andy@ark:~/tmp$ $CC -o hello hello.c
Zig attempted to find the path to native system libc headers by executing this command:
/home/andy/tmp/zigcc -E -Wp,-v -xc /dev/null
Output:
==========
Unable to link against libc: Unable to find libc installation: Zig was not provided with libc installation information, and so it does not know where the libc paths are on the system. Zig attempted to use the system C compiler to find out where the libc paths are, but discovered that Zig is being used as the system C compiler.
See `zig libc --help` for more details.

==========
Unable to link against libc: Unable to find libc installation: libc runtime not found
See `zig libc --help` for more details.