Its most likely a wheel bearing, I had the same problem about a year ago.
Possible wheel bearing going bad.
That noise you're hearing is most likely the wheel bearing. The noise stops because the bearing isn't in motion. As soon as you put the wheel in motion the noise returns due to the bearing being bad. It's a common problem across the board on most Audi's!
Some say that you have to remove the clip-on cover at the back first. Believe to be noise guard.
that is probably your electic fuel pump nothing to worry about
The Audi S4 Avant debuted in 2001. In addition, that year Audi launched the Audi A4 (B6) model. Other Audi models were in production, but these 2 launched in 2001.
Wheel bearing is on its way out.
The 2001 Audi S4 has 30 valves.
The 2001 Audi A4 has 20 valves.
The 2001 Audi TT has 20 valves.
The 2001 Audi A6 has a V6 engine.
The 2001 Audi S8 has 40 valves.