If in doubt, consult a dictionary. The one I have at my desk and the two I checked online give only one pronunciation for router. You can probably guess what it is.
Oh yes, I’d forgotten about this variance in pronunciation.
I pronounce ‘root’ to rhyme with ‘brute’, ‘w00t’, ‘lute’, ‘zoot’, ‘suit’, etc… but not ‘foot’ or ‘hook’. I remember how confused I was when I went to northern Kentucky for work once and met people who pronounced ‘pin’ to rhyme with how I pronounce ‘pen’.
Isn’t English fun?