Take some Saran wrap and tape it down over the drain in your sink just before doing your laundry next time. Watch the Saran Wrap through the entire wash cycle to see whether it becomes concave (indicating suction in the drain pipe) or convex (indicating increased air pressure in the drain pipe) when the washer spins.
I saw a new product commercial on TV. It was a plastic wrap that stuck to itself so well, the you could make "bags" out of the stuff simply by pressing a seam into it with your finger. That stuff might work better than Saran Wrap if you could stick it to the stainless steel just by pressing it against the steel.
What I'm thinking is that if it becomes convex, indicating pressure in your drain piping, it could be because the main drain line from your house is partially clogged, so water is backing up in the drain piping, and possibly compressing air in the drain pipe from your kitchen sink.
Normally, any air pressure in the drain piping will exhaust up the vent pipe on the kitchen sink drain, but often kitchen sinks won't have a vent pipe. They'll often use something called a "Studer Vent" instead. A Studer Vent is nothing more than a spring loaded check valve that allows air in to the drain piping whenever suction created behind draining water threatens to suck the water out of the p-trap under the kitchen sink. By opening to allow air into the drain piping, a Studer Vent ensures that the suction in the drain piping will never be high enough to do that.
Unfortunately, a Studer Vent can't allow a positive air pressure in the drain pipe to exhaust out the vent. As a result, that gurgling noise you hear may actually be bubbles of air bubbling up through you kitchen sink p-trap.
The other possibility is a clogged vent on your kitchen sink drain pipe. (Regular venting this time, not a Studer vent.) If the Saran Wrap goes concave, indication suction in the drain piping, the problem could be a clogged vent PIPE on your kitchen sink drain. When the washer drains, the flow is very turbulent, and bubbles of air in the drain pipe become entrained in the water and carried down the drain pipe with that water, resulting the the creation of a partial vaccuum in your drain piping. If the vent pipe is open, air can enter the drain pipe to fill that vaccuum. However, a clogged vent could result in a partial vaccuum in the drain piping, causing air to be sucked in through your kitchen sink's drain.
Do the Saran Wrap test and let us know the results.