A/C unit calls for "Standard 120v, 3-wire grounding receptacle rated 15 A / 120 VAC /60 Hz. The breaker in the box is rated 10 A. Trying to figure out if the breaker can be switched out with 15 A breaker, can the wiring handle it?

Wow. A 10A breaker. I have never seen one. But no matter.

Frankly, I think your question suggests you would be well advised to read some DIY basic literature before diving into a hazardous project.

To the question: 14 gauge conductors are 0.0641 inches (1.628 mm) diameter, and can be measured with a caliper, with the power off.* Or go/no-go type stamped sheet metal gauges might be used. Or a scrap piece of Romex (NM) cable with marking of AWG size can be visually compared to the conductors.

Heating at the receptacle is a function of current flow and contact resistance, and if the breaker doesn't trip, it is not going to have any effect on receptacle heating. You may not have the needed experience here, but the examination process would be to switch the breaker off, remove the wall plate, test for no power, and then examine the wiring around the receptacle. Wire ends should wrap (cw to the end) around the receptacle terminal screws, which should be tight. If push-in back connections are used, I advise rewiring the receptacle via its screw terminal connections. If the internal metal parts that connect to the plug blades are sloppy, replace the entire receptacle. Note that in most cases the replacement receptacle type has to be of the childproof design.

_______

*Before doing any wiring inspections / manipulation, get a probe type tester that will indicate presence or absence of power to the conductors in a box.

I'd go to the hardware store, buy a foot of 12-gauge wire, and compare it with the wiring in the wall. Perhaps it would be wise to replace the existing outlet with a new one, and make sure the wire is carefully wrapped around the screws and the screws are tightened to make a good connection.

Amperage and Wire Gauge Chart: What Size You Need

Under some conditions, such as the receptacle being dedicated to the air conditioning, the receptacle should be a single outlet type. You wouldn't want someone to plug a vacuum cleaner into the same circuit that the air conditioner is using unless the circuit could handle the combined loads including AC start-up current.

example

My mistake, it's a 15 A breaker! Thanks for your responses. Good idea on the foot of wire by the way. Also good idea to replace outlet, have done a few others in the house, they are long in the tooth and beat up.

The gauge of the wire isn't going to have much to do with the outlet getting warm. It's likely a failing outlet or poor installation so replacing that is a good idea. And buy a good quality outlet - expect to pay $3 to $5 - don't get the cheapest $1 version as linked above!

OP-

Do replace the outlet if you know how to safely do that, and make sure that the wires are attached to it using screwed connections, not simply with those backstab connections where you slide a bare wire into a hole and that's it, as the latter are notorious for poor connections. Also, I'm still curious what the nameplate says the wattage of the AC unit is.

Just replaced the outlet with a very good one and it is not getting warm, that did the trick. Thanks for walking me through this folks , I do appreciate all of your input!