93 4Runner loosing power steering fluid steering gets very stiff and whines when turning. refilling reservoir quiets it down and steers normal. No evidence of a leak. Fluid has bubbles when after adding to warm engine. Bad pump seals? or system leak?

Grinding or whining? A lot of power steering pumps kinda whine when turning. How long has it been since you've changed the power steering fluid? Since 2000??? It looses it viscosity kinda like motor oil. I'd change it first, that's the cheapest and easiest thing to do

Lacking power may mean a burned out pump but whine is caused by air in the pump. A mechanic had me jack up the front end of my van after I refilled the reservoir on the power steering. He said to turn the steering wheel from stop to stop back and forth at least 120 times to work the air out of the pump. Afterwards it seemed to work ok but still had a whine until I got rid of it.

I had a grinding noise on my 1997 Cadillac Deville Limousine and I tried some 'Bar's Leak' Power Steering Fluid that claims to quiet down noisy comopressors, and to thicken and lubricate. So, I tried it 2 days ago, and ran it 100 mi. like it said, and there WAS an improvement in the worn out Power Steering Pump. The cost was like $11 for the product at Walmart, including getting a Turkey Baster to remove the old fluid in the reservoir, that would overflow if I added the whole bottle. I used 1/2 of a bottle and it did seem to improve the sound of the belt driven pump which I gather has worn out splines and bearing inside after 126,000 mi. or so. It certainly was cheaper than spending $500 for a new pump to be installed including evacuating the a/c which I was told was necessary on my particular vehicle.
Hm. $11 last chance try vs. $500. So, I tried the $11 fix and so far it is so much better, that I don't plan to have the power steering pump replaced.

Thank you for the inquiry.
For some reason power steering fluid does get contaminated and needs to be replaced. Worst case the fluid is low and you are hearing gear whine. It's a known issue with this pump to whine when the PSF is contaminated. Needs to be checked and serviced.
Here is a procedure for you Do It Your-self-ers:
You can easily change the fluid yourself, and save a ton of money.

If you follow the lines that go from the steering rack to the cooler located around the radiator area, there's a hose that comes out of the cooler, and connects with a hard line right above the left front frame rail. The connection is a simple spring clamp.

Easy but fairly messy to change. Engine OFF, lift the front of the car to get the front wheels off the ground, put a pan under the hose line, and pulled it loose from the hard line. The bad fluid will drained out, then slowly turn the wheels from side to side a bunch of times to pump out all of the fluid from the rack. When all of the fluid is out, reconnected the line, then fill the fluid reservoir with a high quality Power Transmission Fluid. You can again, turn the wheels side-to-side to get the new fluid into the rack. Then, start the car, again turning the wheels side to side. Top off the reservoir, and should not have any more noise from the power steering pump. It will take a little less than a full quart to completely refill the system.
Kind regards,
TF

Please describe the noise. Is the noise inside the car or beneath the car? Is it a squeak, a clink, or a grind?
If the car has over 100k miles on it, and it makes a hydraulic pump whine when you turn the wheel, look in the reservoir and see if the fluid is still bright red. If the fluid has turned brown, many (including me) have been able to quiet the whine by removing the reservoir and flushing the gunked up screen in the reservoir out with solvent, and then replacing the power steering fluid with fresh fluid.
If you do this yourself, be prepared to dedicate a whole Saturday to the job. If you have any drips from your power steering hoses (common) this will give you the excuse to get in there and replace the leaking hoses while you are at it.

Whining may be normal.
Pulling fluid from reservoir may be normal.
But I would not expect the fluid to go back to the reservior when you stop.
Is it?
If so, you may have some trapped air or something.
Jack the wheels off the ground, and go lock to lock on the steering, several times, to bleed any air out.

Often, noise in the power steering is the result of air trapped in the system. It can also come from contaminated or worn out fluid. If the fluid level in the power steering reservoir is low, the pump will tend to whine and/or chatter. First, check the fluid level. If it is low, there is a leak somewhere. Keep the fluid level up if you must drive it, or the power steering pump will be destroyed. Once you have determined there are no leaks, or repaired any leaks you have have found, the best solution is to remove as much fluid from the system as possible, and then refill it with fresh, clean fluid. There is a product called Lucas power steering stop leak (or somesuch) which is excellent for quieting power steering components and extending their life. Use 1/2qt to 1 qt of regular fluid first, and then the whole bottle of Lucas when refilling. Top off with regular fluid as needed. Be sure to check the fluid level again after letting the engine idle and turning the wheels back and forth several times, as some air will be purged from the system and the level may go down substantially.

When you checked the power steering fluid level were there any indications of excessive foaming? There are three things that would cause a whining noise from the power steering. 1- low fluid (you checked that so that is not the culprit), 2- loose steering drive belt (you should not be able to press down on the belt more that about one inch between the steering pump pulley and the crankshaft pulley), 3- power steering pump bearing worn out.

If its a whining noise, check the power steering fluid level and fill if low, otherwise you may have a pump going out or a loose belt

the suspect part for a power steering whine is the power steering pump, flush (very important) the system and replace the PS pump..