Okay, here's the layout. Kitchen Sink on first floor drains down into basement and then out into the septic tank. Along the way the washing machine is connected to it. Just before the main line goes outside is the 3" cast iron vent stack. The Kitchen sink is not vented before the main 3" vent stack. I recently did my basement over and had all the plumbing redone and really assuming the issue is that the Sink and Washing machine are on the same line is the problem, can I fix it but adding a vent above where the washing machine connects to the main drain? Sorry for any confusion

The sink needs to be vented. If you can't run a stack to the roof then add an AAV. Not legal everywhere so look into air admittence evnts before you install.

EVERY fixture that has a trap MUST have a vent. The gurgle you hear id the washer attempting to vent through the kitchen trap. Both the sink and the washer needs tom be vented.
As Bob pointed out the best way would be to run vent stacks up through the roof. However if that's not possible perhaps a AAV on the sink, (see image) and one on the washer just after the trap would be just as good. I'm a great believer in doing it the easy way. Good luck, Tom