Deaconblues - the cat does not come to work with me every morning, so that's out.
Arthur - I reset the odometer knob each time after I fill up with gas. the sound is not related to the odometer or speedometer cable or so it seems.
Pwood - nope, not the fan motor either.
The purring sound comes on after a few blocks, and only lasts for a few seconds. It only happens after I start the engine.
I just hope this does not become some major trouble later on, which is why I am asking around if someone else has experienced this and might know the cause so I can preempt / avoid it.