I'm eagerly awaiting my (used) copy of this to arrive:
Amazon.com: Professional Linux Kernel Architecture (Wrox Programmer to Programmer) (9780470343432): Wolfgang Mauerer: Books
So I can't really say if it is as good as it looks yet. I'm presuming by "linux API" you mean the kernel, since linux is an OS kernel, so the kernel API would be "the linux API".
However, if you just mean a guide to glibc, the C library used on GNU systems, this is the reference manual:
The GNU C Reference Manual - GNU Project - Free Software Foundation (FSF)
There's html on line or pdf to download. If you want to download the HTML, something like:
wget -r -np -k -p http://www.gnu.org/software/gnu-c-manual/gnu-c-manual.html