You haven't supplied enough information for a definite answer. There are many different models of transfer switch, and several ways said switch could be installed. A picture of the area and specifics about your switch would help. Be aware that there is often a requirement that equipment be installed "in accordance with manufacturer instructions" which if strictly interpreted (not likely) in your case means the switch must go "...18" on either side of the load center...". Will your new transfer switch become the service disconnect? If so, it should be "readily accessable" and if it ends up so low that you almost stand on your head some inspectors might not pass it. You say everything is "crowded together"- so working clearance could be an issue, too.
Speaking of inspections, are you going to have one???- if not you can probably put it where ever you want to. I have seen transfer switches that required a ladder to throw and some 12" off the floor.

And THANK YOU for your decision to use a transfer switch, and not some cobbled together cord, wire or circuit breaker trick.

GULP! This is one of those "pre-wired" multi-circuit transfer switches that requires you to do as much wiring as wiring a much cheaper transfer panel from scratch, right?

[If you're getting the idea that I'm not a big fan of the "pre-wired" flavors of transfer switches, you're spot-on. So, in the interest of full disclosure, I admit to having a bit of prejudice working here. I prefer my spaghetti on a plate, rather than in my electrical panel.]

If I've got the right picture of this transfer switch in my mind's eye, it attaches to the main panel via a short (18"?)nipple of flexible conduit containing in-and-out whips (short wire lengths) for all of the switchable circuits.

While I don't see any problem placing a transfer switch under the panel per se, you may have a problem fitting the conduit from the side of the switch to the bottom of your panel and still have enough wire to reach the bars and terminals.

I dunno. I guess you're about to find out. And, seriously, good luck with your project. (And please don't pay too much attention to my sarcasm. It's a prejudice. I admit it. :-) )

I had forgotten about those things-GAAAH! But they are better than a piece of wire jabbed in the dryer receptacle.

"But they are better than a piece of wire jabbed in the dryer receptacle."

I'm with you on that, Wayne. I second your commending the OP for his or her resolve to put in a proper transfer switch.

When I moved in here, I found an outlet for a dryer out there next to the breaker box. The cable had been disconnected but I figured it had been put there for backfeeding a generator.
The guy across the street is an electrician and told us to backfeed through the air conditioner disconnect box outside.
We have gone too many times without power to the well pump so I decided to do this so I don't kill somebody by rigging things up.
This is the switch.

I think immediately below your panel is a reasonable choice considering what you have there to work with. BTW- you are probably closer to being an "electrician" than the nut across the street.

That's a lot of box fill for that small panel........

Where's the working room for the electrician.....36"x36" clear space ?

I dare say that the "electrician" across the street would not worry about either issue. If a 36" x 36" clear space existed, the original question would not have come up. The addition of pigtail leads and handful of wire nuts used will probably allow the box lid to be reinstalled by less than 3 people.

(Disclaimer- I'm not an electrical inspector, and it would not be possible to pay me enough money to take the job. I already know what the NEC says. I maintain that the resulting install is still likely to be more acceptable than "a piece of wire jabbed in the dryer receptacle".)

The guy across the street works for his father's business in town as an electrician but after he told me to backfeed the generator, I decided I didn't want to trust him with the wiring in my house.
The breaker box is hard to work on with everything crowded into that small space.The water pressure tank is right under it attached to a maze of pvc pipes where you can't put your feet to stand and get close enough to the wall. I found some really large cable hanging down through one of the knockouts on the top that isn't hooked up to anything, just twisted around through the panel. I'll have to call the electric co. to come and tell me what that is for. I'm afraid they'll tell me everything needs to be replaced to get it up to code.