I have a "whirring" noise that I think is from the power steering pump. It's louder than the engine noise when idling. I've seen that some model cars are known for having loud pumps but I've never heard that complaint from an X300. I have a small leak on my rack that will have to be taken care of eventually but this sound is there all the time even after the reservoir is topped off.

 

You might have an airlock in the system.
Try bleeding it by turning full lock to the left and full lock to the right a couple of times with the engine running.

 

Cavitation noise ( air pockets forming as the negative pressure drawing into the pump ) is common from worn out old power steering fluid

Drain refill procedure ( 4 or 5 times ) without removing any hoses

The hoses under the reservoir attach point can leak and go everywhere ( snd can mislead as a rack lead ) and can be resolved by cutting an inch off the hose for fresh clamp material

 

I do have a leak but the pump whirring noise doesn't change when I refill the reservoir. I'll have to look at it closer but I don't think my leak is from the hose. I'm pretty sure its from the rack. I may do the hose trim just as preventative measure anyway.

 

The whirling may dissipate by draining the reservoir ( You can suck out the fill cap ) and filling with fresh , run for a bit cycling the steering to mix old remaining and new fluid

Repeat 4 times as the first mix will be 50 / 50 then the next 75/ 25 abouts

 

PB
Are you really sure it's the power steering pump?
A few days ago, I was helping our local Jaguar club members sort out a deceased club members cars and parts at the request of his widow.
He had 9 cars which had not been started since his demise last year. They ranged from MK4 to X300 including XK120 and E type and all in restored/running order. We were able to eventually start all of the cars
After starting the X300 it sounded like the power steering pump was making a serious noise. There was a lot of geriatric Jag experience present and most of us agreed on the pump.
However, following the noise with a piece of rod used as a stethoscope it was definitely coming from the alternator and not from the pump.
Possibly a bearing or the alternator rotor "poling" on the stator?
Our club members will follow this up with the widow.
I think this is worth a check on your car.

 

I had the front alternator bearing on the X300 give up

After stopping the engine , place hand on stopped pully and feel for heat

If it is the case recommend have the alternator you have just the bearing changed at an alternator shop , least unreliable new alternators other internal parts working these days from reading others

Brushes should be checked for wear

The noise can come from a slipping harmonic balancer with a paint stipe marking test

Water pump bearing noise with a stethoscope test , rod to ear

 

Thanks, I'll have to check that out. It's very possible.

 

On engine startupp before oil warm / fluid viscosity there is a TSB on valve chain rattle with an upgraded chain tensioner located just by the PS pump identified by a large cast arrow on the tensioner


The noise will be from the chains wich are robust so no worry

12-61 (jagrepair.com)