I also have a e60 that the wife drives and her car does the same thing I searched the froum I'm on bimmerfest and came across this a few years ago.
Service Bulletin 18 02 08
Customer may complain of a slight vibration and/or droning noise when the vehicle is idling in park, neutral, drive and reverse when engine is at operating temperature.
Insufficient vibration dampening of the exhaust system.
1.Verify the complaint. Vibration/droning is most prevalent when engine is at operating temperature.
2.Remove the lower bracket for the exhaust system. If droning/vibration is gone, proceed to step three. If vibration/droning still exists with the lower bracket removed; continue troubleshooting exhaust/driveline for other causes or conditions
3.Remove complete bracket assembly and replace it with the bracket (1) PN 18 20 7 521 346 using two new ASA bolts (PN 07 12 9 905 417. Tighten to 21Nm.
Install front pipe bracket (1) PN 18 20 7 521 347 with four rubber bushings (2) (PN 18 20 7 546 579) to the new 346 bracket with two ASA bolts (PN 07 12 9 904 559) and two nuts (PN 07 12 9 904 580). Tighten down to 21Nm
After performing this procedure the vehicle will exhibit the same amount of normal driveline vibration as a comparably equipped E60 with a M54 or N52 engine.
Link to the thread can be found here
. Good luck