How do redirect microphone audio output to headphone output?

Solution 1:

Tested in Ubuntu 16.04 64bit

In a Terminal, type:

alsamixer

Then hit F6 to choose your onboard sound card. Then F4 and set "capture" on 50%, for example. Hit F3, go to the track "loopback" and set it on "enable". Go to track "Mic" and set it on 100%, or any value so you can hear something. If you still hear nothing from your microphone, stay on the track "Mic" and hit the key "m" to unmute the track. That's it!

Works perfectly with my HDA Intel PCH.

Solution 2:

Unless your hardware has a direct monitoring feature and can completely avoid processing the sound through software, there is going to be latency.

It sounds like you are trying to do this on standard built-in audio hardware, which requires software processing to redirect audio, therefore no, there is no way to get rid of the latency.