My 2003 impala has been overheating a lot lately, and the fans nor the thermostat turn on.

Heat sensor may not be kicking in fans, or maybe a blown fuse.
With out a meter to check voltage, you could apply power to fan, and if it doesn't work, it is bad.
If fan is Engine driven, it may be a belt.

Another thought is, if Thermostat is not opening, Radiator may not get hot enough to turn on fan.
What are your sign(s) of overheating, Gauge, losing power, water?

These links should help:

https://www.askmehelpdesk.com/cars-t...tml#post235973

https://www.askmehelpdesk.com/cars-t...ml#post1642158

If it has the 3.4 engine there is a good chance it started from a coolant leak. The lower intake gasket is a known problem on this engine. 3.1 is the same design also. Look under the throttle body for the leak. If the area has a gooey looking substance all over there is a leak. Check under the oil cap and pcv valve for white slude looking buildup. The intake gaseket will leak coolant in to the oil also. If the coolant level got to low it can cause overheating. If it went too long without repair there is a good chance the head gaskets didn't survive. Its about a 5 to 6 hour job at a shop to replace the intake gaskets. A good gasket set plus labor will run $500 plus easily so you want to make sure the headgasket isn't a problem before you start. Get fel-pro permadry gaskets or you WILL be doing the job again in 10 to 15 thousand miles

I also have a 03 impala 3.4 it started overheating on me so I did the first check, check for leaks on all hoses check the color of your exhaust make sure its not white I checked the electrical connections to the fans and temp sensor, all good next keeping it simple I check the relays and fuses all fuses were good so I now think it might be a relay there are three relays that control the fans in this car, your first test in the interior fuse pannel located on the left side of the dashboard there is a relay labeled retained accsry pwr relay, remove it and replace one of the fan relays that match it witch are located in a pannel mounted to the passenger strut tower when you remove the cover you will see two are the same replace one of those 3604 relays with the one you took from fuse pannel start engine if you turn on AC and no fan kicks on swap relays put the original one back in and put the one you took from pannel in other side run it again still no fans take the balck center relay out, get a piece of wire speaker wire will work cut it to about 4" long strip 1/4" from both ends put your key in ignition turn to on position don't start take your jumper wire you just made and in the socket where that black relay was put one end of the wire in slot C6and the other end to B5 if the fans turn on that relay is bad this worked for me it cost me $19.05 for the relay hopefully that's going to be your quick fix to