I have a 1983 Raleigh Gran Prix for my son that has alloy Araya 27 X 1 1/4" wheels. I also have a 82 Trek 610 for my daughter with alloy 27" wheels (the brochure for that year lists Rigida 13-20 as the wheel for them but the labels are worn off so I don't know what they are).

The wheels on the Trek are visibly narrower than the Raleigh's and are fitted with Bontrager 27 X 1 1/8" tires. I would like to find a narrower set of wheels for the Raleigh (actually my son would like a narrower set) but don't know what to look for. I'm not comfortable trying to fit the Bontragers on the Araya rims because they just seem too wide for the tire.

Any suggestions?

Pararacer makes 27 x 1 1/4, 27 x 1 1/8, 27 x 1.The 1 inch are hard to find sometimes but the 1 1/8 are everywhere.They also make a TG(tour guard) tire,with a built in liner.They are a nice tire for the money.I have been running the TG's for about 3 years,no complaints.

I know I can get the tires but what I'm looking for is the narrower rim first. I like the Bontrager's on the Trek.

Some of the alloy 27" rims I come across are 1 1/8", I don't think smaller widths are common. EBay or harriscyclery.com probably have some new sets of wheels starting at $100 or so. Make sure they're alloy, sometimes they're not very specific on this point.

Narrow 27" rims are hard to come by these days, and were probably nearly as difficult in the 80's. I did manage to snag a pair of NOS Araya 27x1 rims a while back that I built up with some Shimano HF hubs. If it wasn't for the crazy price to ship full wheels I'd consider offering to sell them, but I'm out of wheel boxes and they'd probably cost at least $30 just in shipping to get them to you on top of what I'd have to ask for them due to my own investment in them.

Sun CR18 and Sun M-13II are both available at https://sheldonbrown.com/harris/rims/630.html

Find a donor bike. I picked up a bike that had Ukiah rims that are 27 - 1 1/8. FWIW, they were on a touring bike! I swapped them with another bike that had 27 1/4 rims of similar vintage.

If you have the patience, you can find a bike, swap the wheels, then resell the bike. Buy right and you should make a little on the swap.

Thanks for the advice. I'll keep my eyes open. I'm not in a big hurry and can be patient since my son is 12 and just starting to ride.

The Bontragers may go on the Araya's just fine. Measure the interior width and look up the minimum tire width for it in this chart. 1 1/8" is ~29mm so if the interior width is 19mm or less, you should be fine.

https://sheldonbrown.com/tire-sizing.html#width

Before you mount the Bontrager's or any other tire on the Arayas, check the bead on the inside of the rim. Is it a hook, a buldge or is it flat, relying on gum wall and steel bead of the tire to keep them on the rim. If the inside of the Arayas are flat/smooth, maybe you want to just leave the tires that are on there already. If you've got a hook bead or buldge, then I don't see why the 1 1/8 tires won't fit. I've done it with no problem.
As for finding narrow 27" rims, as has been said, they're not common. I have a set that came with my 87 Nishiki Custom Sport. I'm building another set from two narrow front wheels that I found over the last two years, one an Araya and one is a Rigida. I'm converting the Rigida to a rear wheel with a hub and spokes from another damaged 27 inch rim so that I'll have a set.
If you can convert the frame to 700C, you can accomplish your goal much faster as old, freewheel hub 700C are not as hard to find.

Panaracer Pasela's are on sale at Nashbar right now.

Harris Cyclery still lists a narrow 27", 630mm rim as available, Sun M13II.

https://www.sheldonbrown.com/harris/rims/630.html

Also here is Sheldon Brown's article on tire sizing. It includes a section on rim width versus tire width considerations.

https://sheldonbrown.com/tire-sizing.html

This is the other option I've tossed around as well. I'd like to keep it original with 27" but it may be easier to put on some long reach brakes and go to 700C and, thus, open up more possibilities. If I went this route I'd keep the freewheel and hubs on new wheels. Heck, I've even got three sets of Mavic GL330 tubular rims with Campy hubs available - now that could be interesting on a Grand Prix.

The rims on the Raleigh Marathon I just acquired are marked Araya 1-1/4" but they have 1-1/8" tires on them. If 1-1/8 is thin enough, you could try them on the existing rims. If they don't seem to fit right, then worry about getting narrower rims.

You need to measure the rims. I have in my garage several sets of 27x1 1/4 wheels that range from 15 mm to 21 mm between the beads. The narrowest are Weinmann concave, and the widest are some that I got from Nashbar. Even the 15mm wide Weinmann has 27x1 1/4 stamped into it. A 1 1/8 tire on the Nashbar rim is barely wider than the rim, but it stays on. If you look at Sheldon's article, you will see that there is a lot of leeway for rim width vs, tire width.

The Sun rims I mentioned are 22mm and 19.5mm outer width. Is that not narrow enough?

https://sheldonbrown.com/harris/rims.html $40 bucks apiece for the 19.5mm M-13II rims.

metalchef87 has some Araya A20 27x1's he had on the trade thread not long ago, anodized brown laced to Suzue hubs, also anodized brown...