I have a similar problem with my 530i manual (the M30 engined car). There is a whirring sort of noise when the car is cold which diminishes as the engine warms up. It is far more noticeable in lower gears - a sort of rrrr rrrr sound.
It is engine related since it is there whether the clutch is in or out.
One mechanic suggested it could be a worn spigot bearing. In case my American friends have a different name for this it is the bearing that sits in the end of the crankshaft and supports the input shaft which goes into the gearbox.

Hope that helps! I would be grateful for anyone's views. My gearbox is due for an inspection/overhaul but the transmission specialist couldnt do the job because he knew that if he took the exhaust off to take the 'box out, it would fall apart on him! Once I have a new exhaust in place I can think about going back - he will check this bearing and the clutch at the same time.

Sometimes feels like I need a second job to support this old girl...