Centos 6 requires GLIBC 2.12 however a lot of new programs you would want to compile may need a newer glibc. You can't remove the old glibc since the whole OS is based on it but you can install the updated glibc alongside it and do an export pointing to your updated GLIBC.
mkdir ~/glibc_install; cd ~/glibc_install
wget http://ftp.gnu.org/gnu/glibc/glibc-2.14.tar.gz........