I've read a recent study that suggests MINOR colds don't have to interfere with training, and that the duration of the cold is unaffected one way or the other.
Influenza isn't a common cold, however, but a potentially serious viral infection, if that's what you actually have. I'll defer to those here with professional medical training, but I would think that your body's resources need to be concentrated on fighting the flu, which will sap your strength.
I've only had the flu in my youth (been taking flu shots for 30 years), but remember being wasted for a while afterwards. And when I had pneumonia at age 38 (and misdiagnosed for weeks by incompetent Army doctors before it was treated), I was weak as a kitten for a good 6 months following it.
Serious diseases really tax your whole body, and the flu is one of them. I would of course see a doctor, and focus on rest and everything else you're told to do. That's the best way to quickly return to your training.