I have a Rheem gas furnace and Carrier A/C. The furnace blower (which moves the air through the vent system) runs fine when switched-ON via the thermostat however after the AC turns ON for awhile, then the furnace blower stops (after various period of time). When this happens the blower does not work even when switched-ON. The A/C continues to run.

If I shut off the AC & Furnace via the circuit breaker and leave it off for a few hours everything comes back on until the AC turns ON, then the furnace blower stops again.

The fan blower works fine in heating mode.

What could the problem be ?

When it happens again, jump G to R at the thermostat or heater, or just jump it. It will force the fan to continually run. If the fan continues to run, then it's not the thermostat.

It's likely the blower relay.

In heat, the furnace controls the fan, usually on low speed. In AC or FAN modes, the thermostat controls the fan on high speed.

Lots of different scenereos depending on model design.

My thermostat has no "R", I now have a current jumber from Rc to Rh.
Should I ADD a jumber from "G" to "Rc" - or REPLACE the current jumber from Rc to Rh (with "G" to "Rc"?

My thermostat has no "R" - does have "Rc" and "Rh".
Thermostat now has current jumber from Rc to Rh.
Should I ADD a jumber from "G" to "Rc" - or REPLACE the current jumber from Rc to Rh (with "G" to "Rc"?

Leave Rc and Rh jumper in place - Call this terminal R.

Add a jumper from G to (Rc+Rh).

Aside: G only works with Rc, but Rc and Rh must be connected together in your system.