It has two pronunciations just like the word "route". The only difference is that you add an 'r' sound at the end.
Everything has multiple pronunciations, that doesn't make them all correct. While the word "route" has 2 proper English pronunciations (rout or root), the word "router" does not follow the same suit. The correct English pronunciation is, and only is, Rau-Ter.
JXIPlatform no when I say "Rau-Ter" I'm not saying ROTOR...
R, Ow!, Ter!
Like somebody else said, like Cow Turd, but replace the C with an R and drop the D.
Break open a few dictionaries all you will find is:
Pronunciation: \ˈrau̇-tər\