A unlike the older bmw models most models up through 1999 the auxiliary cooling fan on the later models is not just controlled by a coolant temperature.
Bmw e39 aux fan relay.
The switches turn on the fan at low speed and then high speed as the coolant temperature rises.
Your problem is either the temperature sensor or the fan relay.
So i am thinking that maybe all that needs to be replaced is the auxiliary fan switch aka temperature sensor part 13 62 1 433 077 which controls the operation of the auxiliary fan.
It shows the components of the circuit as simplified shapes and the aptitude and signal friends together with the devices.
E39 auxiliary fan wiring diagram wiring diagram is a simplified up to standard pictorial representation of an electrical circuit.
Fuse box in passenger compartment bmw e39.
Tools needed was a 1 4 ratchet 8mm 10mm 13mm sockets.
The temperature sensor screws into the back of the radiator on the reservoir side.
On all bmw models up through the late 90s the auxiliary or main fan is controlled by one or two thermostatic switches that are mounted into the radiator side or bottom tanks.
Switches motors relays fuses wiring for bmw 5 series e39 1996 2003.
For the bmw e39 1996 1997 1998 1999 2000 2001 2002 2003 model year.